Former No 10 spin doctor Coulson, who was forced to resign as Prime Minister David Cameron's director of communications over the scandal, was found guilty of plotting to hack phones while he was editor of the News of the World.
Mr Cameron acknowledged that people would be "concerned" at Coulson having worked for him both as leader of the opposition and Prime Minister, but stressed that there had been no complaints about his work at No10.
The Prime Minister said he had asked the former News of the World editor whether he knew about phone hacking before he was hired, and received assurances which have now turned out to be false.
Speaking in the Cabinet Room at 10 Downing Street, Mr Cameron said: "I take full responsibility for employing Andy Coulson.
"I did so on the basis of undertakings I was given by him about phone hacking and those turned out not to be the case.
"I always said that if they turned out to be wrong, I would make a full and frank apology and I do that today.
"I am extremely sorry that I employed him. It was the wrong decision and I am very clear about that."
Coulson now faces the possibility of jail following the high-profile trial at the Old Bailey.
However, the jury of eight women and three men cleared ex-News International chief executive Rebekah Brooks of all charges.
Retired managing editor Stuart Kuttner was also cleared of being part of a conspiracy dating back to 2000 and spanning six years.
Brooks's former personal assistant Cheryl Carter was cleared of conspiracy to pervert the course of justice. Brooks's husband Charlie and former NI director of security Mark Hanna were also cleared of perverting the course of justice.
But the jury is still considering further charges against Coulson and former NotW royal editor Clive Goodman of conspiring to commit misconduct in a public office by paying police officers for two royal directories.
Mrs Brooks appeared to try to keep her emotions in check when the first of the not guilty verdicts was announced, simply nodding towards the jury with a slight smile.
Dressed in a white blouse, she appeared to hold the hand of her former personal assistant Cheryl Carter as she was cleared.
The pair looked at each other in the dock after the not guilty verdict against Ms Carter was given to the court.
On her other side was her husband Charlie, who stood with his hands folded in front of him as he was cleared. His wife brushed him comfortingly as the verdict was given, while he stood staring straight ahead.
Coulson, dressed in a dark suit, showed no immediate reaction to the guilty verdict, standing with his hands behind his back and clenching his jaw while looking forward.
A few moments later he took a deep breath in and out.
Mr Cameron is expected to make a "profound apology" following Coulson's conviction.
Coulson was recruited by George Osborne to head up the Tory media operation within months of resigning as News of the World editor in January 2007.
When Mr Cameron entered Downing Street the former journalist took on duties heading up the Number 10 spin operation, quitting shortly before he was arrested over the phone-hacking scandal.
A spokesman for the Prime Minister said Mr Cameron "entirely" stood by his words to parliament in July 2011, when he told MPs: "I have said very clearly that if it turns out Andy Coulson knew about the hacking at the News of the World he will not only have lied to me but he will have lied to the police, to a select committee, to the Press Complaints Commission and, of course, perjured himself in a court of law."
Rebekah and Charlie Brooks left the Old Bailey just before 12.45pm.
Greeted by a throng of media, the couple left court holding hands and made no comment as they got into a waiting black cab.
The partial verdicts were delivered on the jury's eighth day of deliberations and the 138th day of the trial.
Ex-No 10 spin doctor Coulson, 46, had denied all the charges as had Goodman, 56, of Addlestone, Surrey.
Brooks, 46, was cleared of hacking, misconduct in a public office for allegedly signing off payments to a Sun journalist's "number one military contact" between 2004 and 2012, conspiracy to pervert the course of justice and perverting the course of justice.
Carter, 50, was cleared of perverting the course of justice by removing seven boxes from the NI archive just days before she was arrested in 2011.
Racehorse trainer Charlie Brooks, 52, and Hanna, 51, were cleared of perverting the course of justice around the time of police searches in July 2011.
The prosecution had alleged that because of the the sheer scale of phone hacking at the NotW, Brooks, Coulson and Kuttner must have known what was going on while they were in charge.
After Clive Goodman and Glenn Mulcaire were convicted of phone hacking in 2006, all three bosses were alleged to have been involved in a cover up to keep the full extent of phone hacking secret.
Three more senior ex-NotW journalists have since admitted involvement in tasking the private detective - news editor James Weatherup, chief reporter Neville Thurlbeck, and assistant editor Greg Miskiw.
And reporter Dan Evans, who was caught hacking the phone of designer Kelly Hoppen in 2009, became a star witness by implicating 10 former colleagues, including Coulson.
It was Brooks who was the editor when Mulcaire got his first annual contract with the paper for £92,000 in 2001.
She was still in charge in 2002 when Mulcaire hacked the phone of murdered schoolgirl Milly Dowler - an act which led to the eventual downfall of the Sunday tabloid.
Even though she was on holiday in Dubai at the time, the prosecution said her on-off love affair with her deputy Coulson meant they shared such confidences.
Kuttner was also implicated by the fact he told Surrey Police about the Milly voicemail himself, the prosecution alleged.
After Coulson took over the editorship in 2003 and Brooks went to the Sun, hacking really took off at the NotW, the court heard.
Goodman, 56, said it was going on on an "industrial scale", while Evans told jurors even the "office cat" knew.
A host of politicians, sports people, celebrities, and members of the royal family were targeted by NotW staff and Mulcaire.
The then Kate Middleton was hacked 155 times, Prince William was hacked 35 times and Prince Harry nine times.
Goodman claimed his former boss had personally sanctioned his Alexander Project - a separate £500 a week deal with Mulcaire to target three royal aides between autumn 2005 and spring 2006.
And Evans said he showed Coulson a voicemail from the actress Sienna Miller to Bond star Daniel Craig indicating an affair.
An email about Calum Best in which Coulson wrote "do his phone" was, the prosecution said, a direct hacking order from the boss, although there was no firm evidence that the footballer's son was a victim.
Coulson admitted he knew about a hacked voicemail from former home secretary David Blunkett declaring his love for Spectator publisher Kimberly Quinn in August 2004 - but insisted that was the only one.
The prosecution said phone records and the fact the Sun named Ms Quinn the day after the NotW published its exclusive meant Brooks knew too.
Kuttner was also implicated in December 2004 when he rang journalist Simon Hoggart to put it to him that he was having an affair with Ms Quinn as well, the court heard.
Despite bosses knowing about the Milly and Blunkett phone hacks, they neither investigated nor disciplined anyone involved but instead agreed for it to continue, the prosecution asserted.
But Brooks, Kuttner and Coulson all denied they were involved in the hacking conspiracy.
Brooks said she was unaware of Mulcaire and what he was doing.
The only published story proved to come from hacking while she was editor came from the Milly voicemail and she was "shocked" to find out about it in 2011.
Coulson said he did not know about Mulcaire either and assumed Thurlbeck had hacked Ms Quinn himself.
His first reaction was to ban his chief reporter from investigating the story any further, but he was later persuaded it was justified in the public interest - a decision he admitted was a mistake.
Kuttner said he was an old fashioned journalist who believed in getting stories through traditional means.
The 74-year-old Fleet Street veteran signed off payments to sources and planned annual budgets at the NotW but said he did not cook the books to hide Mulcaire's annual salary which rose to £105,000.
Coulson is also accused of conspiring with Goodman to commit misconduct in a public office.
Emails were shown to the court in which Goodman asked Coulson to authorise paying police officers for two royal directories.
But Goodman said he was prone to exaggerate his sources' importance and Coulson said he never actually believed they were public officials.
The sources have never been identified.
Brooks faced a similar charge by agreeing to pay a Sun journalist's "number one military contact" on 11 occasions while editor of the daily paper.
She denied knowing it was a public official and said she trusted the reporter concerned.
When the hacking scandal re-emerged in July 2011, Brooks was accused of trying to hide or destroy evidence from police.
The ex-News International (NI) chief executive was charged with conspiring with her personal assistant Cheryl Carter to pervert the course of justice.
Carter had removed seven boxes marked as Brooks's notebooks from the NI archive on the day the closure of the NotW was announced.
But she said they mainly contained her notebooks which she binned and anything belonging to her boss was returned to the office.
Around the time of her arrest on July 17 2011, the prosecution said there was an attempt by Brooks, her husband and Hanna to pervert the course of justice.
Bags of property were found stashed in the underground car park of the Brooks's London flat by a cleaner and handed to police.
But Mr Brooks said the assorted items, which included lesbian porn DVDs as well as computer equipment, were all his and had nothing to do with the case.
Former News of the World managing editor Mr Kuttner today gave his "enduring thanks" to his legal team as he was cleared of being part of the phone hacking conspiracy.
As Rebekah Brooks was cleared of all charges at the Old Bailey, Mr Kuttner was also cleared of being part of a conspiracy dating back to 2000 and spanning six years.
Speaking outside the court, he told reporters: "It is clear to me at this point that this is not the moment to make long statements or to go into great detail.
"But what I do want to say is the diligence, the dedication, and perhaps above all the passion of my lawyers over the last three years has been extraordinary, most remarkable, and it is to them that I owe the huge and enduring thanks for the result, the unanimous verdict of the jury today. Thank you."