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The newspaper published a prominent public apology on page two of its print edition today, but the move – which came two days after the parent company issued a similar statement – showed no sign of calming the fury of those affected.
Lawyers for Sienna Miller, one of the most prominent admitted victims – said she had not accepted any offer of a settlement over the "outrageous violation of her privacy".
The Former MP George Galloway, who says he has seen proof that his phone was hacked, dismissed the apology as "a cynical attempt to protect the company's chief executive", Rebekah Brooks.
In its statement the News of the World said the hacking "should should not have happened. It was and remains unacceptable." Speaking of the victims, it said: "Here today, we publicly and unreservedly apologise to all such individuals."
The cabinet minister Danny Alexander described the hacking as "a very serious scandal", adding that the court cases and police investigation "must go forward".
"It's outrageous that people have had their voicemails hacked into, seemingly a large number of people," the chief secretary to the Treasury said on BBC1's Andrew Marr Show.
Alexander repeated previous government statements that the affair would have no bearing on News Corp's attempt to take over BSkyB, an issue he said was "completely separate".
The shadow Welsh secretary, Peter Hain, called for a "full and proper public investigation" and said the police investigation had been "tardy".
The Observer revealed that Rupert Murdoch tried to persuade Gordon Brown as prime minister to get Labour to back off on the affair.
A former minister in Brown's cabinet said there was evidence that Murdoch relayed messages to Brown via a third party, urging him to help take the political heat out of the row, which he felt was in danger of damaging his company.
News International's apology confined direct blame to "a News of the World journalist and a private investigator working for the paper" – the former royal editor Clive Goodman and private investigator Glenn Mulcaire, both of whom were jailed in 2007.
No mention was made of the arrests last week of its chief reporter, Neville Thurlbeck, and assistant editor Ian Edmondson, who has been sacked.
John Whittingdale, chair of the culture, media and sport select committee, told the Sunday Telegraph that more arrests were like. "I think there'll be more arrests because as I understand it there's more evidence that points at other people being involved," he said.
The News International statement said an unreserved apology, admission of liability, and compensation was being offered "in certain civil cases that meet specific criteria", but that the company would "continue to contest cases what we believe are without merit or where we are not responsible".
The News of the World apology said: "Evidence has recently come to light which supports some of these claims. We have written to relevant individuals to accept liability in these civil cases and to apologise unreservedly, and will do the same to any other individual where evidence shows their claims to be justifiable."
