The pleasant spa town of Bad Kreuznach, astride the River Nahe in the rolling farmland of southwestern Germany, was until a few years ago best known for its American army base and its fine local Riesling. But the Americans have since moved away, the economic blow only partly softened by the expansion of a local Michelin plant. This year's grape harvest has been threatened by downpours of near-biblical proportions.

And now, Bad Kreuznach finds itself at the centre of one of the most bizarre, high-profile murder mysteries in the country's history - the search for an apparent serial killer whom police and prosecutors call, simply, 'The Woman Without a Face'. They have no fingerprints to go on. No witnesses. No description. What they do have is a trail of DNA, now stretching back 15 years and across three countries - as well as a grisly new reason to put a face to her double helix. A case that had for years been gnawingly disturbing, yet still fairly obscure, has leapt on to the front pages of German newspapers. For it appears now that the mystery woman may not only be a killer, but a cop-killer.

In April 2007, Michèle Kiesewetter, a 22-year-old policewoman from an elite drugs squad, was taking a lunch break with a colleague in their BMW patrol car in Heilbronn, nearly 100 miles from Bad Kreuznach. Two people climbed into the back seat and shot the officers from behind, killing the woman and seriously injuring her 25-year-old partner. The assailants struck so quickly their victims had not even drawn their weapons.

As dozens of green-uniformed officers, carrying a smiling portrait of their murdered colleague, paraded at her funeral, the case shocked and moved the country. It also sparked one of the largest criminal investigations in German history - with results that at first puzzled, then stunned, the investigators.

The only clue was microscopic traces of DNA, found on the centre console and the rear passenger seat of the BMW. And when the samples were finally fully secured, tested and compared with Germany's central crime database nearly three months later, there was an extraordinary match - from two quite different murder scenes stretching back a decade and a half. The 'Woman Without a Face' had, it seemed, struck again.

If nothing else, the frenzy of media coverage that followed did at least give the mystery woman a name. 'The Phantom of Heilbronn', the headline writers soon dubbed her, as newspapers, magazines and documentary-makers chronicled the police efforts to hunt down their elusive suspect. And the more the police probed, the more matches they turned up, not only with a string of further crimes across southern Germany but, as the DNA call went out across Europe, with nearly a dozen break-ins and vehicle thefts across the border in Austria and France.

Most galling of all for the investigators, her DNA signature has continued to turn up at new crime sites since Heilbronn - most bizarrely, perhaps, a few months ago when the corpses of three Georgian car dealers were trawled from a river near Heppenheim, south of Frankfurt. Two men were jailed for the killing, an Iraqi and a Somali. In the Iraqi suspect's battered old Ford, forensic officers found traces of the same DNA found in the police car in Heilbronn. But how did it get there? Who is the woman whose genetic calling card has been found at more than 20 scenes of theft, assault and murder hundreds of miles and more than a dozen years apart? If their Iraqi suspect can help, police say, he's not telling.

More than 100 police and prosecutors in three separate teams across Germany, backed by DNA analysis from the BKA in Wiesbaden (Germany's equivalent of the FBI) and by officers in Austria and France are now involved in an increasingly frantic effort to answer those questions. But nowhere is the sense of determination - and growing frustration - felt more keenly than in a large, pale-yellow building in the centre of Bad Kreuznach.

With its long, cavernous hallways and quiet cubbyhole offices, it has the feel of a Fifties insurance office, or a firm of actuaries. In fact, it is the headquarters of the prosecutors' office of the state of Rheinland-Pfalz. And it was here, 15 years ago, that the search for the Woman Without a Face began - with a DNA sample on the rim of a brightly painted teacup.

The cup belonged to a 62-year-old woman in the nearby town of Idar-Oberstein, favoured by tourists for the Church of the Rock perched on the hills outside and by businessmen as one of the leading gem-cutting centres in Europe. On 23 May 1993, a neighbour who had knocked on her door and got no answer phoned the police. When they arrived, they found Lieselotte Schlenger dead, strangled by a strand of wire taken from a bouquet of flowers in her sitting room.

'The only clue was the DNA,' recalls Günter Horn, the boyish-looking 44-year-old prosecutor in charge of unravelling a mystery that has since reached far beyond that first killing. Dozens of potential witnesses were interviewed at the time, but no one had seen or heard anything suspicious. After a while, the case was all but forgotten, set aside in the increasingly vain hope that fresh evidence might one day turn up. Perhaps, police figured, the woman might at some point be arrested for another, less serious crime, and their puzzle piece would fit.

But eight years later, in 2001, Horn says as he thumbs through two now-bulging pink cardboard files on the case, came a much more alarming match. In Freiburg, in the far southwestern corner of Germany, a 61-year-old antiques dealer was found dead - again, strangled. The DNA at the scene was identical to that at the Idar-Oberstein murder.

It next appeared five months later, just a few dozen miles from Bad Kreuznach, on a discarded heroin syringe. The soiled needle was turned in to police by a distraught woman whose seven-year-old son had unsuspectingly stepped on it in a playground in the town of Gerolstein, not far from the border with Belgium. And in the increasingly fraught police efforts to unravel the mystery since the policewoman was killed in Heilbronn, checking the evidence against nearly half a million records held at the BKA in Wiesbaden, a dizzying mosaic of other DNA matches has emerged - and with it the beginning of a portrait of their quarry, and the company she keeps.

On the night of 24 October 2001, two weeks after the syringe was found, a caravan was burgled on the outskirts of Mainz, not far from Bad Kreuznach. DNA taken from an abandoned biscuit outside matched the Woman Without a Face. On New Year's Day in 2003, there was a break-in at an office in Dietzenbach, seven miles outside of Frankfurt. Her DNA was there, too. In December 2003, a car was stolen in Heilbronn. It was later abandoned, and when it was tested, her DNA was found on the petrol cap. In Karlsruhe in 2005, there was a late-night robbery at a bar. Her DNA was found on two beer bottles and an empty wine glass.

The following year, across the French border in Besançon, it turned up again - on a toy pistol used in a robbery. And it turned up, too, in nearly a dozen break-ins at shops and offices, and several car thefts, in Austria. At some of the crime scenes, the genetic print of presumed accomplices was also found, though according to Horn 'it was never the same ones'. At least three men have been arrested - from Slovakia, Serbia and Moldova. But again, if they know anything about the Phantom of Heilbronn, they're not saying.

And on the afternoon of 6 May 2005, she - or traces of her - reappeared much closer to Bad Kreuznach, at an attempted murder in the cathedral city of Worms. A member of the local gypsy community turned a 7.65-calibre pistol on his brother, and police promptly arrested the gunman - only to find, after forensic checks in Wiesbaden, that the mystery woman's DNA was on one of the bullets.

'But there are still no witnesses, and no other evidence,' says Horn. 'All of us on the various teams talk to each other two or three times a month. We meet, we email, but mostly we wait for another report saying the same DNA has turned up.'

The most recent, he says with a resigned sigh, came 'just a few months ago - from a fishing lodge in a little town called Saarhölzbach, in Saarland, near the Luxembourg border. Someone came in at night and sneaked up behind the woman who was in charge of the cleaning staff, struck her hard on the back of the head, stole 300 euros, and left.

'Our suspect's DNA was found in the room - not on the woman who was robbed, and who never saw the attacker, but in the room ... Another clue, I guess. But to be honest, we're still no nearer to knowing who she is.'

It is not for lack of trying. A €100,000 reward is now on offer. Saliva swabs have been taken, too, from hundreds of women in southern Germany, France, even Belgium and Italy.

The testing strategy has been based on the hints - guesses, really, Horn acknowledges - from which the investigators have been struggling to build a coherent picture of the woman whose trail began on the rim of a teacup 15 years ago.

The syringe suggests drug use. The apparent randomness of the break-ins, the small amounts often taken, point to targets of opportunity and a desperate need for cash. The fact that many of the crime scenes have been in or around Horn's patch suggest that the Woman Without a Face is 'still in southern Germany, still nearby'. Yet the geographical range of some of the crimes, the bizarre links with the murdered Georgians and the bullet used in the gypsy feud in Worms have convinced the police that she may have ties with one of the loosely linked groups and communities who move back and forth across Europe's increasingly porous frontiers.

Horn, pointing to the evidence from the Austrian burglary sites, speculates that she may be part of an 'organised group of burglars from Eastern Europe'. But a colleague quickly interrupts: 'Even in jail, these people don't talk.'

The DNA testers have taken samples from nearly 3,000 'homeless women, women who are drug users, women who have a record of a serious crime', Horn says. But so far there has been no match. He adds that he is not at all surprised by the growing, gruesome fascination of the case for the media. 'There are the murders. And the mystery.' And, of course, what Horn calls the allure of 'the modern equivalent of fingerprints - DNA.'

DNA stands for deoxyribonucleic acid, which carries the genetic instructions from which all human beings are made, and it earned the scientists James Watson and Francis Crick a Nobel Prize for their work in identifying how it was structured.

But it was another British academic, the Leicester University geneticist Sir Alec Jeffreys, who paved the way for the marriage between genetic science and crime-solving. In the Eighties, Jeffreys pioneered 'DNA profiling' - based on generating a digital code, with a shrinkingly small probability of being found in another person, from the individual pattern in which brief portions of our DNA repeat themselves.

The technique was first used to crack a major crime in 1988, when Jeffreys was called in to help solve a pair of shocking cases, three years apart. Two teenage girls in the Leicestershire village of Narborough, Dawn Ashworth and Lynda Mann, had been sexually assaulted and murdered. A young man was already in custody and had reportedly confessed to one of the killings. But DNA profiling soon established that a local baker, named Colin Pitchfork, was in fact the murderer.

It was a breakthrough which, in the two decades since, has transformed the way crime is investigated, prosecuted and, of course, written about. DNA 'evidence' has dominated media coverage of countless high-profile crimes, most dramatically the murder of a number of prostitutes in Ipswich in 2006, and the disappearance of Madeleine McCann last year. Even crime fiction has changed beyond recognition. Sherlock Holmes has given way to the likes of Kay Scarpetta, the DNA-savvy forensic scientist created by the best-selling American crime writer Patricia Cornwell.

So far, the one thing that DNA evidence has unequivocally revealed about the presumed murderer Günter Horn has spent years hoping to find is that she is a woman. Some of the limitations of what else can be gleaned from the rim of a teacup, an empty beer bottle, a biscuit crumble, a toy gun or a real bullet are peculiarly German. In the shadow of a Holocaust in which genetic pseudo-science became a handmaiden to genocide, there are strict limitations on DNA records held in Wiesbaden. Only the full details of individuals convicted of serious crime can be added to the database. And so far, at least, the Woman Without a Face is also a woman without a criminal record.

There are limits, too, in the aftermath of Hitler's glorification of his blue-eyed, blond-haired 'master race', on what kind of scientific questions can be asked of the DNA samples that have linked the Woman Without a Face to the far-flung crime scenes. Race and ethnicity are off-limits, as are eye and hair colour.

Horn is particularly quick to bat away suggestions raised in the wake of the Heilbronn killing, which was not far from a gypsy community, and the clan shooting in Worms, that the mystery woman may have ties to the gypsy, or Romany, community, which was targeted during the Holocaust, alongside the similarly 'non-Aryan' Jews.

'There are lots of people and communities who move around,' he says, before adding that the real problem - his real frustration - is that there are other, more serious limitations to the almost magical power now sometimes attributed to DNA. He points to rumours in Austria, where limits on DNA testing are less stringent, suggesting that she may have blonde hair and blue eyes. 'But the fact is that there would be a significant percentage of error in any such conclusion ... And in a population of 80 million people, that error would be measured in hundreds of thousands of people.'

It is a message of caution echoed at Europe's leading DNA crime laboratory, the Forensic Science Service (FSS) in Birmingham, and by Jeffreys himself. After Portuguese police first talked up their later discredited 'DNA evidence', suggesting Madeleine McCann's parents might have been involved in her disappearance, Jeffreys was quoted as saying that even the FSS relies on a genetic database with insufficient 'markers' fully to rule out the possibility of error.

A spokesperson at the FSS adds that Horn is right to be especially sceptical of the value of reports suggesting that Germany's mystery killer might be blonde and blue-eyed. 'It is true,' she says, 'that there is a test for identifying redheads, but only because so few of the population have red hair.' But even a test like that can't 'identify' a criminal. And in cases where there is 'nothing else to go on', genetic profiling - particularly since the technique works by using repetitive 'junk' DNA rather than the genetically encoding sequences - can provide 'only intelligence, only an inference'.

Yes, says the spokesperson, DNA can be an enormously powerful tool in crime investigations, especially in matching a culprit to a series of crimes, or in reheating 'cold cases'. But in the end it is only 'one tool' alongside many others - not least what Günter Horn describes as good, old-fashioned 'police work'.

In Bad Kreuznach, and across Germany, that police work continues. Horn fully expects another round of calls, emails and reports in the days ahead from the other teams at work on what has become the most intractable case he and his colleagues have ever handled.

The pressure for a solution is steadily building, stoked by a stream of breathlessly worded reports in the media and by a growing array of speculation and conspiracy theories in a town that still awaits a solution to the murder that started the mystery 15 years ago. 'Of course I'm following the case of The Phantom. We all are,' remarks Christof Braun, a local taxi driver. 'Every few weeks, it seems, there's a story that this DNA has been found at a crime scene somewhere else. Who knows? Maybe the Phantom is even dead, and some other criminal is leaving traces of her DNA to fool the police!'

Horn, for his part, suspects the solution to the string of murders and other crimes, when it comes, will begin with the addition of a remarkably ordinary face to the still-unfocused picture assembled from the genetic traces she has left so widely and for so long. 'The Phantom' is not, he is certain, some 'criminal genius' who is cleverly outwitting the police and prosecutors of Germany and much of Europe. 'She leaves no fingerprints because she wears gloves,' he says. 'That is pretty standard.' There are no witnesses, presumably, because she steals in, she strikes, and she leaves.

Horn and his colleagues also recognise that even if they do finally match the DNA that has turned up at crime scenes around Europe with its owner, that will be only the beginning, not the end, of their real work. They will have to establish definitively how and why the DNA - and its owner - got there; what role she played, particularly at crime scenes where the DNA tests suggest other intruders were with her; and of course build a case that will stand up in a court of law.

And if he and his colleagues remain confident that they will eventually get their woman, Horn is frank when asked how long that might take. 'Maybe another five years,' he says, smiling. 'Maybe 10 years. Or maybe tomorrow.'

So is he waiting for the Woman Without a Face to slip up - to leave a clue more useful, if less tantalising, than her DNA? For a witness? A tip-off? 'Maybe all of those things,' Horn replies. 'And in the end, just a bit of luck.'