PM David Cameron defended his decision to employ Mr Coulson as his aide.
Late on Friday, an unnamed 63-year-old became the third man to be arrested by police investigating phone hacking.
He was arrested at an address in Surrey on suspicion of corruption. Police are currently searching the property.
Earlier, former NoW royal editor Clive Goodman, 53, who was jailed in 2007 for phone hacking, was released on bail following his arrest on suspicion of corruption
Mr Coulson, 43, was arrested at 1030 BST on Friday by detectives investigating allegations of hacking the phones of various people in the news, and of corruption.
Remit of police investigations
- Operation Weeting - investigating phone hacking or intrusion into the private lives of hundreds of people. They aim to contact all those whose personal details were found in documents seized in 2006
- Operation Elveden - investigating alleged police corruption. Documents handed over by News International on 20 June were assessed by police as including "information relating to alleged inappropriate payments to a small number of officers".
Mr Cameron said of Mr Coulson, his former communications chief: "I became friends with him and I think he did his job for me in a very effective way. He became a friend and he is a friend."
Meanwhile, it has been revealed that Rebekah Brooks, head of the NoW's parent firm News International, is no longer heading the firm's own inquiry into the scandal.
She told News International staff in an e-mail that those carrying out the investigation would now report to Joel Klein, a US-based senior executive at the company's owner, News Corp.
On Friday, Mrs Brooks held a meeting with NoW staff at its headquarters in Wapping.
A source present at the talks told the BBC she had informed staff they would eventually understand why the Sunday tabloid had to close.
News International has said it is shutting the NoW after this Sunday's edition following a spate of fresh revelations.
The 168-year-old tabloid is accused of hacking into phones of crime victims, celebrities and politicians. Police have identified 4,000 possible targets.
In other developments:
- Mr Cameron says the Press Complaints Commission (PCC) should be scrapped and replaced with an entirely new system
- Prosecutors have asked Strathclyde Police to examine specific claims of phone hacking in Scotland by the NoW
- A number of charities - including the Salvation Army, Care International and the RSPCA - have rejected an offer to advertise for free in the final edition of the NoW on Sunday
He said a judge-led inquiry would look into "why did the first police investigation fail so abysmally; what exactly was going on at the News of the World and what was going on at other newspapers".
A second inquiry would examine the ethics and culture of the press, he added.
'Cathartic moment' The prime minister also questioned the tenability of Mrs Brooks as News International chief executive considering she was editor of the NoW at the time murdered schoolgirl Milly Dowler's phone was being hacked.
The controversy has raised questions about the proposed takeover of satellite broadcaster BSkyB by Rupert Murdoch's News Corporation, publisher of the NoW.
And broadcasting regulator Ofcom has now written to the chairman of the Commons culture committee highlighting the watchdog's duty to ensure that anyone holding a broadcasting licence is a "fit and proper" person to do so.
The letter says "in considering whether any licensee remains a fit and proper person to hold broadcasting licences Ofcom will consider any relevant conduct of those who manage and control such a licence".
Mr Coulson, who was the paper's editor at the time, said he took "ultimate responsibility" for the scandal but insisted he was unaware of any phone hacking by his journalists.
Discussing his decision to employ Mr Coulson as his director of communications in 2007, the prime minister said: "I decided to give him a second chance but the second chance didn't work. The decision to hire him was mine and mine alone."
Mr Cameron admitted politicians were to blame for "turning a blind eye" to bad practices in journalism but said the controversy had led to a "cathartic moment" for both the media and politicians.
Afterwards the Labour leader Ed Miliband said the prime minister "clearly still doesn't get it".
He said: "His wholly unconvincing answers of what he knew and when he knew it about Mr Coulson's activities undermine his ability to lead the change that Britain needs."