Milly Dowler's family have been told by police that the News of the World may have hacked the phone of their daughter Milly while she was missing.

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The news comes just 11 days after Levi Bellfield was convicted of abducting and murdering the 13-year-old in 2002.

It is an extraordinary development in the phone hacking scandal, code-named Operation Weeting. The allegation that a private investigator working for the News of the World tried to hack in to the phones of the Dowler family at a time of such distress throws a whole new light on the possible extent of the hacking.

The Dowler's family lawyer says The News of the World's former private investigator Glenn Mulcaire has been accused of illegally accessing Milly's phone messages after she went missing in March 2002. There will probably have been numerous private and clearly desperate messages on her phone from her parents at the time.

Scotland Yard officers contacted Sally and Bob Dowler about the allegations in April, a month before Levi Bellfield went on trial for her murder, solicitor Mark Lewis said.

"The fact that they were prepared to act in such a heinous way that could have jeopardised the police investigation and give them false hope is despicable." Mark Lewis, Dowler family lawyer

Mr Lewis, from London-based Taylor Hampton Solicitors, said: "Sally and Bob Dowler have been through so much grief and trauma without further distressing revelations to them regarding the loss of their daughter.

"It is distress heaped upon tragedy to learn that the News of the World had no humanity at such a terrible time.

"The fact that they were prepared to act in such a heinous way that could have jeopardised the police investigation and give them false hope is despicable."

Interviewed on Channel 4 News, the Labour MP Tom Watson said that once the police investigation had been completed the Government should order a public inquiry.

Mr Watson is a member of the Culture, Media and Sport Select Committee, which has held an inquiry into the allegations. He said there was "a lot more to come out", including, he believed, allegations involving phone-hacking in the case of the Soham murders of schoolgirls Holly Wells and Jessica Chapman.

"Since I have been involved in this inquiry, there are a number of whistleblowers that I have spoken to and I believe there is a strong suspicion that one of the Soham parents was targeted by Glenn Mulcaire," he added.

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The News of the World's publishers, News International, said in a statement: "We have been co-operating fully with Operation Weeting since our voluntary disclosure in January restarted the investigation into illegal voicemail interception.

"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."

The issue of the Dowler family's phones being hacked into was raised earlier this year but, remarkably, one of Scotland Yard's most senior officers (and formerly in charge of the phone hacking investigation) claimed never to have heard about it.

Tom Watson put the following question to John Yates, at the time Acting Deputy Commissioner at Scotland Yard, at a hearing of the Culture, Media and Sport committee on 24 March:

Mr Watson: Third, we know from the Information Commissioner’s inquiry into the convicted private investigator Steve Whittamore that private information about a family member of Milly Dowler was obtained. If it transpires from the review of the Mulcaire evidence that, when Sky News were broadcasting it round the clock, Glenn Mulcaire was instructed to hack the phones of the family members of children killed at Soham, would that warrant an adequate use of police resources to investigate it?

John Yates: I am sure it would, but that is the first I have ever heard of that aspect.

The Metropolitan Police would not comment on the matter today.

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Earlier, Mr Watson called in the Commons for a Government statement on the issue.

"As well as being a despicable and evil act that will shock parents up and down the land to the very core, it also strongly suggests that Parliament was misled in the press standards inquiry held by the DCMS (Department for Culture, Media and Sport) Select Committee in 2010," he said.

And Labour's former Deputy Prime Minister, Lord Prescott, said on Twitter that, following the latest allegations against the News of the World, he was writing to the Culture Secretary, Jeremy Hunt, urging him to block the takeover of BSkyB by Rupert Murdoch's News Corp.