Media playback is unsupported on your device Media caption Mr Cameron said police need to pursue their inquiry in the "most vigorous way they can"

Claims that murdered teenager Milly Dowler's mobile telephone was hacked by a private investigator working for the News of the World (NoW) are "truly dreadful", the prime minister has said.

Police are to meet executives from the paper to discuss allegations that Glenn Mulcaire hacked into Milly's voicemail when she was missing in 2002.

Labour has demanded a public inquiry into the claims.

NoW parent firm News International said the development was of "great concern".

Speaking during a news conference in Afghanistan, David Cameron said the allegations were "quite shocking - that someone could do this, actually knowing that the police were trying to find this person and trying to find out what had happened".

He added: "If they are true, this is a truly dreadful act and a truly dreadful situation."

The Guardian has claimed Mulcaire intercepted messages left by relatives for the 13-year-old and said the NoW deleted some messages it had already listened to when the voicemail facility became full.

A lawyer for the Dowler family, Mark Lewis, said the alleged hacking dated from a time when the NoW had been under the editorship of Rebekah Brooks - now News International's chief executive.

In a statement, he said: "The fact that they were prepared to act in such a heinous way that could have jeopardised the police investigation and gave them false hope is despicable."

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Scotland Yard launched Operation Weeting in January after new claims of phone-hacking at the NoW emerged.

Mr Lewis said the Metropolitan Police contacted the Dowlers in April this year about the hacking and also told them that their own phones had been targeted.

They would be seeking a claim for damages against the NoW, he added.

Milly went missing in March 2002 near her home in Walton-on-Thames, Surrey and her remains were found in remote woodland at Yateley Heath in Hampshire six months later.

Nightclub doorman Levi Bellfield was convicted of the murder last month.

Met Police Commissioner Sir Paul Stephenson declined to comment on the ongoing police inquiry into phone-hacking but said he was "appalled" by the latest allegations.

Surrey Police, which conducted the original inquiry into Milly's disappearance, said it would be inappropriate to comment in light of the Met investigation.

Earlier, Mr Cameron said police needed to pursue their criminal investigation into the case in the most "vigorous" way they can.

"The police in our country are quite rightly independent, they should feel that they should investigate this without any fear, without favour, without any worry about where the evidence should lead," he said.

But questioned on whether the latest developments would affect the bid by the News of the World's parent company News Corporation to take a majority stake in BSkyB, Mr Cameron said he would not intervene as that was a decision being taken by Culture Secretary Jeremy Hunt.

Media playback is unsupported on your device Media caption Mark Lewis said hearing about the hacking allegations gave him "goosebumps"

The Guardian also alleges that the NoW employed another private investigator, Steve Whittamore, to illegally obtain ex-directory numbers for families called Dowler living in Walton-on-Thames.

A News International spokesman said it had been co-operating fully with the police inquiry into hacking since News International's "voluntary disclosure in January restarted the investigation into illegal voicemail interception".

He said: "This particular case is clearly a development of great concern and we will be conducting our own inquiries as a result.

"We will obviously co-operate fully with any police request on this should we be asked."

Nick Davies, the special correspondent from the Guardian who wrote the story, told the BBC: "This is one of the very few episodes that happened when she [Rebekah Brooks] was editing the paper, and she's clearly going to have to answer some questions about what she knew about what was going on."

Labour leader Ed Miliband described the alleged hacking as "absolutely awful" and said the police inquiry had to "get to the bottom of who was responsible for this and who was complicit in it".

He said British journalism has had "one its lowest days".

He added: Mrs Brooks should "examine her conscience" and consider her position although the issue of phone-hacking at the News of the World went "beyond one individual".

Image caption Schoolgirl Milly Dowler went missing nine years ago

Paul Connew, who was deputy editor of News of the World in the late 1990s, told BBC Radio 4's Today programme the latest claims "raise a lot of challenging questions as to who knew what and when".

But BBC business editor Robert Peston said Mrs Brooks is expected to tell staff she is "deeply shocked" about the latest claims and insists she knew nothing about phone-hacking at the newspaper.

She has spoken to Mr Murdoch and is under no pressure to resign from her role at News International, he added.

The BBC's political editor Nick Robinson said the allegations had "changed the character, if not the nature, of the hacking saga" and "moved many in Westminster who previously regarded the story as a question of interest only to those excited by media ethics or the privacy of celebrities".

In January, the High Court will hear claims from five test cases involving public figures who say their phones were hacked into.

They arise out of the disclosure of information by the Metropolitan Police relating to material forfeited by Mulcaire.

He and former NoW royal editor Clive Goodman were jailed in 2007 over hacking into the phones of members of the royal household. They pleaded guilty to conspiracy to intercept telephone calls between 1 November 2005 and 9 August 2006.

Five other journalists have been arrested as part of the current police investigation.