Newspaper 'Hacked Milly's Mobile.'

The parents of murdered Walton-on-Thames schoolgirl Milly Dowler are suing the News of the World over claims a private investigator working for the newspaper hacked into her phone after she went missing.

Bob and Sally Dowler said they had been given ``false hope'' their daughter could still be alive after voicemail messages on the phone were deleted in the days after she vanished.

Solicitor Mark Lewis said the ``heinous'' and ``despicable'' actions could have jeopardised the police investigation.

Scotland Yard officers contacted Mr and Mrs Dowler about the hacking allegations in April, a month before Levi Bellfield went on trial for her murder.

News International, the publisher of the newspaper, said the allegations were of ``great concern'' and said it would be conducting its own inquiry.

Private investigator Glenn Mulcaire is alleged to have illegally accessed Milly's phone messages after she was abducted by Bellfield as she walked home from school in Walton-on-Thames, Surrey, in March 2002.

The voicemail messages were allegedly deleted to make room for more after Milly's inbox became full.

Mr Lewis, from London-based Taylor Hampton Solicitors, said Mr and Mrs Dowler were pursuing a claim for damages against the News of the World and the Dowlers had been told their own phones were targeted, as well as that of their daughter.

Prime Minister David Cameron condemned the alleged hacking of Milly's mobile telephone as a ``truly dreadful act'', and said the claims were ``quite, quite shocking''.

Bellfield, 43, was given a second whole life jail term last month for 13-year-old Milly's murder.

Ms Brooks spoke out in a message to News International staff after David Cameron condemned the alleged hacking as a ``truly dreadful act''.

The Prime Minister, who found himself being quizzed about the story during a trip to Afghanistan, said he found the claims ``quite, quite shocking''.

As the scandal grew, the Press Complaints Commission, a keystone of the system of press self-regulation in the UK, accused the News of the World of lying during an inquiry into hacking allegations.

PCC chairman Baroness Buscombe told the BBC's Daily Politics show: ``We didn't have the evidence. I am the regulator but there is only so much we can do when people are lying to us.

``We know now that I was not being given the truth by the News of the World. Who knows if there are other newspapers that have lied.''

In her statement Ms Brooks said she wanted to address staff ``as a matter of urgency'' in light of the new claims.

She said: ``We were all appalled and shocked when we heard about these allegations yesterday.

``I have to tell you that I am sickened that these events are alleged to have happened. Not just because I was editor of the News of the World at the time, but if the accusations are true, the devastating effect on Milly Dowler's family is unforgivable.

``Our first priority must be to establish the full facts behind these claims.

``I have written to Mr and Mrs Dowler this morning to assure them News International will vigorously pursue the truth and that they will be the first to be informed of the outcome of our investigation.

``Our lawyers have also written to their solicitor Mark Lewis to ask him to show us any of the evidence he has so we can swiftly take the appropriate action.

``At the moment we only know what we have read.

``Since 2006, when the Metropolitan Police Service (MPS) seized the documentation from the private investigator Glen Mulcaire, News International has had no visibility on the evidence available.

``The process of discovery is complicated. The MPS first present relevant documents to potential victims. We only see the evidence much later during the legal process.''

She said NI has offered the MPS its full co-operation ``to establish the veracity of these fresh allegations'' and had written to the Chief Constable of Surrey, where Milly lived, with the same offer.

She went on: ``I am determined that News International does everything it can to co-operate fully and pro-actively with the MPS, as we have been doing for some time, to verify the facts so we can respond in a robust and proper way.

``It is almost too horrific to believe that a professional journalist or even a freelance inquiry agent working on behalf of a member of the News of the World staff could behave in this way.

``If the allegations are proved to be true then I can promise the strongest possible action will be taken as this company will not tolerate such disgraceful behaviour.

``I hope that you all realise it is inconceivable that I knew or worse, sanctioned these appalling allegations.''

She added: ``Although these difficult times will continue for many months ahead, I want you to know that News International will pursue the facts with vigour and integrity.

``I am aware of the speculation about my position. Therefore it is important you all know that as chief executive, I am determined to lead the company to ensure we do the right thing and resolve these serious issues.

``We will face up to the mistakes and wrongdoing of the past and we will do our utmost to see that justice is done and those culpable will be punished.''