London (CNN) -- Journalists hacked into the phone messages of a missing girl, deleting some to make space for more and thus giving her parents hope she was still alive when she was dead, the parents' lawyer told CNN on Tuesday.

"The family are completely horrified. They thought this was all over" after the disappearance of Milly Dowler in 2002 and the conviction of a man for her murder this year, lawyer Mark Lewis said.

But in April, police told the Dowlers that journalists had hacked into their phones and those of their daughter, he said.

The accusation is the latest twist in a long-running scandal involving media baron Rupert Murdoch's flagship Sunday British tabloid newspaper, the News of the World.

The paper has apologized for hacking into the voicemails of celebrities and politicians, paying compensation to actress Sienna Miller and offering money to others.

But the Dowler case is the first time the newspaper is accused of interfering with a police investigation.

British Prime Minister David Cameron on Tuesday called the allegations "quite shocking" and urged the police to investigate thoroughly.

A former top tabloid journalist Tuesday called the allegations a significant change in the course of the scandal.

"Before we had hacking of celebrities and politicians, but here we have the interference of a murder inquiry involving a 13-year-old girl," said Roy Greenslade, a former editor of the Daily Mirror and an assistant editor of the Sun, a sister paper of News of The World.

"Outside of the political elite in Britain, this story has had little traction until now. The allegations about the hacking of Milly Dowler's phone has the potential to change that," Greenslade said.

Police Monday declined to say whether Milly Dowler was among the victims of phone hacking.

Meanwhile a private investigator working for News of the World who was jailed over another phone hacking case, in 2007, issued a statement to the UK's Guardian newspaper apologizing for the "hurt" caused by his actions.

Glenn Mulcaire did not mention hacking Milly Dowler's phone but said he had never intended to interfere in any police inquiry.

He said he had faced "relentless pressure" for results at the paper, and that while he knew they were pushing ethical boundaries, he had not understood they were breaking the law.

Executives from Murdoch's News International met with British police Tuesday over the claims, the company told CNN.

One of the top company officials told staff later that it was "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."

Rebekah Brooks, who was editor of the paper at the time of the alleged hacking and is now chief executive of News International, told staff she was "sickened that these events are alleged to have happened."

She said she had contacted the Dowlers on Tuesday "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."

Brooks asked Lewis, the Dowler lawyer, to show the company any evidence it had "so we can swiftly take the appropriate action."

She said News International was cooperating with police investigations.

CNN obtained a copy of the e-mail Brooks sent to staff on Tuesday. News International confirmed it was genuine.

Brooks acknowledges in the message that there is speculation she might resign, but said she was "determined to lead the company to ensure we do the right thing and resolve these serious issues."

The tumult may have an impact on the newspaper's advertising revenues.

Ford Britain said it was withdrawing its advertising from News of the World until there was an outcome from its investigation.

"Ford is a company which cares about standards of behaviour of its own people and those it deals with externally," the carmaker said in a statement.

A spokesman for T-Mobile said the firm was reviewing its advertising position with the News of the World, while supermarket chain Tesco said the latest allegations would "cause huge distress to a family which has suffered enough" and said it was awaiting the result of the police inquiry.

At least five people have been arrested in connection with phone hacking investigations this year since a new investigation, Operation Weeting, was launched in January.

Mulcaire and a journalist working for the News of the World were sent to prison in 2007 for hacking into the voicemails of royal staff in an earlier investigation.

Police launched the new investigation this year in response to widespread complaints from politicians, celebrities and other high-profile figures who fear they have been targets.

News International has apologized for unspecified cases of phone hacking. They say they have been cooperating with police since the new investigation was launched in January.

The Sunday tabloid newspaper in April offered compensation and "apologized unreservedly" for the "unacceptable" hacking. It did not name the victims.

News International owns the News of the World, plus the Sun, the Times and the Sunday Times in Britain.

Murdoch's media empire also encompasses Fox News, the Wall Street Journal, the New York Post and Harper Collins publishers.

CNN's Jonathan Wald, Dan Rivers, J.P. Landry, Peter Wilkinson and Antonia Mortensen contributed to this report.