Francesca Hilton, who died on Monday after suffering a fatal heart attack in a downmarket Los Angeles coffee shop

Paris Hilton was her great-niece, Elizabeth Taylor had been her sister-in-law, Richard Burton, Sean Connery and Frank Sinatra were among her mother’s lovers, and she grew up in a Bel Air mansion once owned by billionaire Howard Hughes and Elvis Presley.

But it was in a downmarket 24-hour coffee shop that former Hollywood princess Francesca Hilton, 67, suffered a fatal heart attack on Monday and later died in hospital.

Friends of Francesca, the only child of screen siren Zsa Zsa Gabor, claim she had felt depressed for weeks and kept repeating that she wanted to die.

For Francesca was down and out, living sometimes in a car and sometimes in a sleazy Hollywood motel.

As the ghost-writer of her mother Zsa Zsa’s autobiography, One Lifetime Is Not Enough, I first met Francesca in the early Eighties. She had last contacted me two months ago.

For years she had been working on her own autobiography, tentatively entitled Hotels, Diamonds And Me, and there was talk that she might want to work with me on the book, though nothing came of it.

It’s a shame, because Francesca Hilton had one hell of a story to tell.

The only child born to Zsa Zsa — one of Hungarian sisters the Gabors, who specialised in marrying rich men and collecting magnificent jewellery along the way — Francesca would appear to have been born with a silver spoon in her mouth.

Her blonde bombshell mother had starred in the 1952 hit film Moulin Rouge and then made a name for herself as a TV personality, uttering such bon mots as: ‘I’m a great housekeeper. When I divorce a man, I always keep his house.’ But she wasn’t worth a fortune.

As for Conrad Hilton, the hotel magnate whose name Francesca bore, he never even mentioned her in his 1957 memoir and left her just $100,000 of his fortune, worth about $6 billion (nearly £4 billion) in today’s money, when he died in 1979.

In the decades that followed, Francesca devoted her life and any money she had to fighting her father’s will, but lost resoundingly — and even, under a quirk of California law, forfeited the original $100,000.

Those who knew her believe her fight wasn’t just for money but to establish without doubt that Conrad was her father — which he once disputed to her but not to Zsa Zsa, who said that if he did ‘I would have killed him’.

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Friends of Francesca, pictured, the only child of screen siren Zsa Zsa Gabor, claim she had felt depressed for weeks and kept repeating that she wanted to die

Conrad and Zsa Zsa were separated at the time of Francesca’s conception — they divorced in 1947 — and although Zsa Zsa assured me that Conrad visited her one day in 1946, raped her and that Francesca was the result, her story is in doubt.

Back in the Seventies, the right-hand woman of Conrad Hilton’s son, Barron, told me that Francesca’s father was not Conrad but the actor George Sanders. The resemblance to Sanders is startling.

To the end, Francesca considered herself a Hilton. But if she was on shaky ground there, anyone who saw her with her mother felt that she didn’t resemble her much, either.

With her matchless allure, which had captivated billionaire Paul Getty (who wanted to marry her), the Duke of Marlborough and two U.S. Presidents, John F. Kennedy and Richard Nixon, Zsa Zsa was dedicated to her beauty, glamour and sex drive (she once confided to me that she liked a man to be rough in bed).

As a child, Francesca adored her mother, but Zsa Zsa did her best to turn her little girl into a live-in photo opportunity, throwing her lavish birthday parties to which she invited reporters from Life magazine.

Zsa Zsa with her daughter, left, in the 1950s and right in the 1960s. As Francesca grew up, though, Zsa Zsa was dismayed when she balked against wearing make-up and losing weight, and rejected the glamorous Gabor image

As Francesca grew up, though, Zsa Zsa was dismayed when she balked against wearing make-up and losing weight, and rejected the glamorous Gabor image. Instead, Francesca was honest, funny, cynical and bright, so that when young reality TV star Paris Hilton hit the headlines, she remarked: ‘I’m the original Hilton heiress. I’m older, wiser, smarter. And I’m damned wider . . .’

Although Francesca didn’t grow up to be a man-eater like her mother, she had a crush on singer Kris Kristofferson, dated Peter Sellers and once infuriated Joan Collins by flirting openly with Joan’s then husband, Ron Kass.

As a child, Francesca adored her mother, but Zsa Zsa did her best to turn her little girl into a live-in photo opportunity

Yet love, happiness and career success eluded her.

Conrad Hilton refused to pay her an allowance and wouldn’t even let her stay one night on his luxurious estate.

Only once did he grant her a temporary role in his hotel empire, as a receptionist at the Beverly Hilton, where guests flatly refused to believe she was a member of his family.

After that, Francesca tried valiantly to become a professional photographer, then an actress, and managed to play a few bit parts in films.

Her last career was as a stand-up comedienne, in an act that consisted of poking fun at Zsa Zsa (‘my mother and I, we’re the best of friends now we’re the same age’), her heritage and, above all, herself.

Towards the end of her life, though, Francesca had very little to laugh about.

After marrying and divorcing eight men, in 1986 Zsa Zsa married her final husband. Born Hans Robert Lichtenberg, a policeman’s son from Germany, her new spouse — who purported to be 27 years her junior — had been adopted as an adult by the elderly German princess Marie Auguste Von Anhalt.

Now going by the title of Frederic Prinz Von Anhalt, which is his legal right, he paid a contact $5,000 to introduce him to Zsa Zsa, then married her.

Zsa Zsa and Frederic fought constantly, but she was proud of their bawdy sex life and even prouder that she — ‘a little girl from Budapest’ — bore one of the most ancient titles in Europe.

Frederic fought a dogged battle to keep Francesca away from Zsa Zsa. On the few occasions when he allowed them to meet, Francesca found her mother his virtual prisoner.

Stricken with dementia and with one leg amputated, Zsa Zsa was able only to squeeze Francesca’s hand to let her know she knew who she was.

‘She once called me The Brat, so that’s how I announce myself,’ Francesca said wryly. ‘I say: “Mother, the Brat’s here.” ’

Zsa Zsa Gabor pictured with her husband Frederic Prinz von Anhalt in 2011. Frederic fought a dogged battle to keep Francesca away from Zsa Zsa

But Francesca was far from being a brat. She was a tortured woman who grew up with very little love, grappling with a mother more concerned with herself than with her child.

‘Do I have breakfast with my mother? Of course not. She always has it in bed,’ she once said, without bitterness.

Since her death was announced, Frederic Von Anhalt has said openly that he has no intention of letting his wife, now 97, know that her only daughter is dead.