Media playback is unsupported on your device Media caption Ed Miliband urges David Cameron to appoint a judge to lead an inquiry into phone hacking allegations

David Cameron has promised to set up a public inquiry into phone hacking at the News of the World.

The UK prime minister said claims that the voicemail of murdered schoolgirl Milly Dowler had been hacked, with some messages deleted, were "disgusting".

But he said an inquiry must wait until police investigations were over - Labour says it should be set up sooner.

Meanwhile News Corporation chief Rupert Murdoch said the allegations were "deplorable and unacceptable".

But he stood by News International chief executive Rebekah Brooks, who was editor of the News of the World at the time, and has faced calls for her resignation.

Mr Murdoch, the head of News Corporation, the newspaper's parent company, said in a statement: "I have made clear that our company must fully and proactively cooperate with the police in all investigations and that is exactly what News International has been doing and will continue to do under Rebekah Brooks' leadership."

He added that he was "committed to addressing these issues fully".

'Revolted'

It is claimed that 13-year-old Milly Dowler's voicemail was hacked by an investigator working for the News of the World after she disappeared near her home in Surrey in 2002 .

This follows allegations that dozens of politicians and celebrities, including actor Hugh Grant and former Deputy Prime Minister Lord Prescott, were also targeted.

Analysis Ed Miliband went to the Commons to demand an inquiry that he felt the prime minister would be reluctant to concede. But David Cameron took the wind from his sails by raising the possibility not only of one inquiry but two: one into into media standards, another on how the police handled the initial hacking revelations. It is not clear when the PM decided to back an inquiry but those close to him say he saw the current allegations - in particular the hacking of murdered schoolgirl Milly Dowler's phone - as being "particularly serious". But Mr Cameron still faces two difficulties raised by Mr Miliband. First, he has been reluctant to change the timescale for the potential takeover of BSkyB by the Murdoch empire because "this is a quasi-judicial process". Expect pressure to be applied to get the whole deal referred back to the Competition Commission. Second, Andy Coulson - David Cameron's former communications chief - has returned to prominence over allegations of police payments. Despite today's announcement, on the explosive issue of hacking the touchpaper remains alight.

It emerged on Wednesday night that Chancellor George Osborne has been informed by police that his name and home phone number had been found in notes kept by private investigator Glenn Mulcaire and the former News of the World reporter Clive Goodman. A spokesman for the chancellor said there was no suggestion his phone had been hacked.

Amid noisy scenes at Prime Minister's Questions in the House of Commons, Mr Cameron said: "We do need to have an inquiry, possibly inquiries, into what has happened.

"We are no longer talking here about politicians and celebrities. We are talking about murder victims, potentially terrorist victims, having their phones hacked into.

"It is absolutely disgusting, what has taken place, and I think everyone in this House and indeed this country will be revolted by what they have heard and what they have seen on their television screens."

But he added that an inquiry could not happen yet, as there was a "major police investigation under way".

Mr Miliband told MPs it was "possible for the prime minister to start the process now".

The Labour leader also questioned Mr Cameron's decision to hire another former News of the World editor, Andy Coulson, as his director of communications after he resigned from the paper in 2007 over the phone hacking scandal, calling it a "catastrophic error of judgement".

Mr Coulson resigned in January saying claims about phone hacking were making it impossible to do his job.

Mr Miliband urged Mr Cameron to back his call for Mrs Brooks to resign from her current job as chief executive of News International - the UK arm of News Corporation.

But the prime minister said it was important to "let the police do their work" before making claims about the conduct of individuals.

'Constitutional issue'

The prime minister's spokeswoman told the BBC there could even be two inquiries into phone hacking - one into the police handling of the original investigation in the middle of the last decade, and one into the actions of the media.

Or, alternatively, there could be one all-encompassing inquiry, led by a judge.

MPs have also been holding an emergency debate on phone hacking.

Media playback is unsupported on your device Media caption Chris Bryant: "Frankly this is News International doing 'yeah but, no but, yeah but'"

Labour's Chris Bryant questioned the role of the Metropolitan Police during the earlier investigation into hacking, and the information officers had given ministers and others.

He said: "I think a lot of lies have been told to a lot of people. When police officers tell lies or, at best, half-truths to ministers of the Crown... that's a major constitutional issue for us to face."

Fellow Labour MP Tom Watson called for News International chairman James Murdoch to resign, adding that he and Mrs Brooks had to "accept their culpability and they will have to face the full force of the law".

Attorney General Dominic Grieve said the government would do all it could to "progress matters further" regarding an inquiry.

But some MPs - including some Conservatives - also urged a "pause" in any decision on whether News Corporation should be able to take full ownership of BSkyB.

As revelations involving the News of the World continue to emerge, families of victims of the 7 July bombings in 2005 have complained that they may have had their phones hacked and police investigating the claims have contacted the parents of murdered schoolgirls Holly Wells and Jessica Chapman.

On Tuesday News International passed e-mails to the police which appear to show that payments to police officers had been authorised by Mr Coulson when he was News of the World editor.

Mrs Brooks has said the claims of hacking Milly's phone are "almost too horrific to believe" and that it is "inconceivable" that she knew about them during her time in charge of the paper.