British Prime Minister David Cameron led condemnation yesterday of “truly dreadful” allegations that a British tabloid hacked the voicemail of a missing teenage girl who was later found murdered.

He said police should pursue their investigation into the claims about the News of the World, Britain’s top-selling Sunday title owned by Rupert Murdoch’s News International, in “the most vigorous way they can”.

However, Mr Cameron made clear that the allegations about Milly Dowler, who disappeared aged 13 in 2002, would not prompt him to intervene in the bid by Murdoch’s News Corp. to take full control of satellite broadcaster BSkyB.

News International chief executive officer Rebekah Brooks said she was “sickened” by the claims and vowed to “vigorously pursue the truth”.

Ms Brooks was editor of the News of the World when Miss Dowler went missing, but she said it was “inconceivable that I knew or worse, sanctioned these appalling allegations”, and rejected calls for her to quit.

“I am determined to lead the company to ensure we do the right thing and resolve these serious issues,” she said.

The tabloid has been dogged by claims of phone hacking ever since its royal editor and a private investigator were jailed for the practice in 2007, and a new investigation was launched in January amid a wave of fresh allegations.

But most of the targets have been celebrities and politicians – the paper recently settled a civil suit with actress Sienna Miller – so claims that Mis Dowler’s phone was hacked sparked widespread outrage.

The teenager disappeared in 2002 on her way home from school near London, sparking a high-profile search. Her bones were found six months later, although it was not until last month that her killer Levi Bellfield was convicted.

According to the Guardian, private investigators and journalists listened to increasingly desperate messages on Miss Dowler’s phone left by her parents and friends as the days went by without any word from her.

When her voicemail box became full, they even deleted several messages to make room for new ones – an action that her loved ones and police mistakenly took as proof that Miss Dowler was still alive and using her phone, the report said.

“If they are true, this is a truly dreadful act and a truly dreadful situation. What I’ve read in the papers is quite, quite shocking,” Mr Cameron told a press conference during a visit to Afghanistan.

He said police “should feel they should investigate this without any fear, without any favour, without any worry about where the evidence should lead them. They should pursue this in the most vigorous way that they can”.

However, Mr Cameron said he would not intervene in the government’s deliberations over News Corp.’s highly controversial bid to buy the 61-per cent of BSkyB it does not already own.

“The government, on these processes, is acting in a quasi-judicial way and it is quite right that the Secretary of State for Culture, Media and Sport (Jeremy Hunt) carries out his role in that manner without any interference from anyone else in the government,” he said.

Police refused to comment on the allegations about Miss Dowler, but Ms Brooks said News International would cooperate fully with any investigation, as well as conducting its own internal inquiries.

“I have to tell you that I am sickened that these events are alleged to have happened,” she said in a message to staff, adding that she had written to Miss Dowler’s parents, Sally and Bob, promising to “vigorously pursue the truth”.

She continued: “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.”

The Dowlers’ lawyer, Mark Lewis, earlier said they would be suing the tabloid for its “heinous” and “despicable” behaviour and said Ms Brooks should “do the honourable thing” and quit.

“She should consider her position but this goes well beyond one individual,” added opposition Labour leader Ed Miliband. He called for a public inquiry, and an emergency debate on the issue was set for the House of Commons today.