Search for new No 10 media chief after Coulson quits
The search is under way for a new director of communications at No 10 after Andy Coulson resigned.
Mr Coulson stood down on Friday, saying coverage of the News of the World phone hacking scandal had "made it difficult to give the 110% needed in this role".
He will stay at Downing street for a few weeks while a successor is found.
Alastair Campbell, Tony Blair's former spin doctor, said he thought Mr Coulson "could have withstood a great deal more scrutiny and pressure".
Mr Campbell said he was surprised to hear that Mr Coulson was stepping down.
"His departure - the dramatic nature of it and the coverage it's now getting - has drawn greater attention to the really serious issues in this," said Mr Campbell.
"They are David Cameron's judgement, the role of the News of the World and what appears to have been industrial phone hacking, the BSkyB decision and Jeremy Hunt's handling of that, and the conduct of the police."
Mr Coulson quit as the paper's editor in 2007 saying he took ultimate responsibility for the scandal, but denied knowing about the phone hacking.
In a statement, he said it had been "a privilege and an honour to work for David Cameron for three-and-a-half years" and he was proud of the work he had done.
But he added: "Unfortunately, continued coverage of events connected to my old job at the News of the World has made it difficult for me to give the 110% needed in this role.
"I stand by what I've said about those events but when the spokesman needs a spokesman, it's time to move on."
In a statement Mr Cameron praised him as a "brilliant member of my team".
The prime minister said he was "very sorry" Mr Coulson had decided to resign but he "understood that the continuing pressures on him and his family meant that he felt compelled to do so".
"Andy has told me that the focus on him was impeding his ability to do his job and was starting to prove a distraction for the government," he said.
He denied that it raised questions about his own judgement and said Mr Coulson had "run the Downing Street press office in a professional and competent and good way".
The BBC's political correspondent Mike Sergeant said Mr Coulson had become a "powerful political insider - precisely because he had a feel for the world outside Westminster" and the PM would be looking for a "someone with similar nous and journalistic experience".
A number of current and former editors and correspondents are now being talked about for the role.
Joanna Nadler, who worked in the communications office when John Major was prime minister, said "as far as possible they will want to try to recruit another newspaper person".
"There is a feeling that particularly tabloid journalists are just that little bit sharper and more streetwise if you like," she said.
Mr Coulson was editor of the News of the World in 2007 when its royal editor, Clive Goodman, was jailed for conspiracy to access phone messages. Private investigator Glenn Mulcaire was jailed for six months on the same charge.
A Press Complaints Commission investigation in May 2007 found no evidence that he or anyone else at the paper had been aware of Goodman's activities.
That same month he became Mr Cameron's director of communications.
But pressure has recently mounted on him, amid renewed newspaper investigations into the scale of phone hacking at the Sunday tabloid. Mr Coulson himself was interviewed as a witness by police in November.
In December, Director of Public Prosecutions Keir Starmer said no new charges would be brought in the case, owing to a lack of admissible evidence.
But some public figures are taking civil legal action against the newspaper, and documents disclosed in those cases have led to new developments.
Earlier this month the News of the World suspended its news editor, Ian Edmondson, over allegations of phone hacking in 2005-6, thought to involve the actress Sienna Miller.
Asked about Mr Coulson's resignation, Labour leader Ed Miliband said: "We think he should have gone earlier, he has now done the right thing. I think there are questions about David Cameron's judgement about hanging on to him as long as he did."