Rupert Murdoch has apologised personally to Milly Dowler's family over claims the murdered girl's voicemails were accessed illegally by the News Of The World.
It is alleged the Sunday tabloid hacked into the teenager's mobile after she was abducted in 2002, when Rebekah Brooks was editor.
Messages were deleted to make room for more recordings, it is claimed, and this gave the family false hope that she was still alive.
Mrs Brooks has resigned as News International chief executive after coming under increasing pressure since the allegations about Milly's phone emerged a couple of weeks ago.
Mrs Brooks denies any knowledge of the alleged wrongdoing.
And the scandal has claimed another senior executive at News Corporation.
Les Hinton, who worked for the company for 52 years, has stepped down as chief executive officer of Dow Jones & Company.
He worked at News International from 1995 until 2007 when he told the Culture, Media and Sport Select Committee he had not seen any evidence the hacking had spread beyond one rogue journalist.
Meanwhile, an apology written by Mr Murdoch over the phone hacking scandal will appear in national newspapers on Saturday.
He had called the meeting with the Dowlers and the family's lawyer Mark Lewis at a London hotel.
Speaking afterwards about the Milly claims, Mr Murdoch told reporters: "I was appalled about what had happened and I apologise."
Mr Lewis said: "He [Mr Murdoch] was humbled to give a full and sincere apology to the Dowler family.
"The Dowler family told him that his papers should lead the way to set the standards of honesty and decency in the field, and not what had gone on before.
"At the end of the day, actions are going to speak louder than words."
Mr Lewis also said Mr Murdoch "apologised many times" and "had his head in his hands".
The hacking claims have centred on the now closed-down News Of The World (NOTW), which was published by News International (NI), the UK subsidiary of News Corp, and officers say around 4,000 people could have been targeted.
In his letter of apology, Mr Murdoch writes: "We are deeply sorry for the serious wrongdoing that occurred.
"We are deeply sorry for the hurt suffered by the individuals affected. We regret not acting faster to sort things out.
"I realise that simply apologising is not enough.
"In the coming days, as we take further concrete steps to resolve these issues and make amends for the damage they have caused, you will hear more from us."
Among the hacking allegations are that the NOTW could have illegally accessed the phones of 7/7 victims' families as well as relatives of dead soldiers.
And the FBI has said it is investigating claims News Corp journalists tried to hack into the phones of 9/11 victims.
In a statement Les Hinton said: "I have watched with sorrow from New York as the News of the World story has unfolded.
"I have seen hundreds of news reports of both actual and alleged misconduct during the time I was executive chairman of News International and responsible for the company.
"The pain caused to innocent people is unimaginable."
Mr Murdoch praised Mr Hinton's "extraordinary work" and said: "Les and I have been on a remarkable journey together for more than 52 years.
"That this passage has come to an unexpected end, professionally, not personally, is a matter of much sadness to me."
Earlier Mrs Brooks admitted she had become a "focal point of the debate" and had to quit to avoid distracting attention from her company's efforts to "fix the problems of the past".
The former Sun and NOTW editor, 43, said she was "sorry for what we now know to have taken place".
Her resignation came despite Mr Murdoch describing her as his first priority when he flew to London last Sunday to take charge of the crisis in his media empire.
Meanwhile, Mrs Brooks' successor at the paper, Andy Coulson, stayed as a guest of the Prime Minister at Chequers in March, several weeks after his resignation as David Cameron's communications chief, Sky sources say.
Mr Coulson was arrested last week by police investigating alleged phone hacking at the publication. He was questioned and later released on bail.
Mrs Brooks received hospitality at Chequers twice last year, once in June and again in August.
In recent weeks, the share prices of News Corp, the parent company of NI, and BSkyB have fallen amid the scandal.
News Corp has withdrawn its bid to take full ownership of the broadcaster after the proposed deal was criticised by politicians.
Police are continuing their investigations into alleged hacking and there will be a public inquiry led by judge Lord Justice Leveson.
Meanwhile, NI has denied completely a legal action by Jude Law against The Sun alleging that four articles published in 2005 and 2006 were based on information derived from illegal voicemail interception.
On Friday night the actor issued a statement through his lawyers Atkins Thomson confirming the action.
It said: "News International have put out a statement about a claim brought by Jude Law, which is currently before the court.
"We have no comment to make about this claim which will be considered by the court in due course."
News International said in a statement: "We believe this is a deeply cynical and deliberately mischievous attempt to draw The Sun into the phone-hacking issue.
"The allegations made in this claim have been carefully investigated by our lawyers and the evidence shows that they have no foundation whatsoever."
Law is already taking action against the News of the World and the case will be heard in January at the High Court.
Sky's political editor Adam Boulton said of Mrs Brooks' resignation: "It doesn't change any of News Corp's legal problems.
"It's probably too late for the sacrifices - last week the NOTW and today Rebekah Brooks - to stop any of the inquiries: the criminal inquiries, the public inquiries or the investigations in the United States."
Mrs Brooks is due to appear before MPs on the Commons Culture Committee next Tuesday where she is expected to face tough questions about hacking allegations at the paper, which was closed at the weekend.
Also giving evidence will be Rupert Murdoch and his son News International chairman James Murdoch , who both bowed to pressure after previously saying they were unavailable to attend on that day.
Labour leader Ed Miliband had been calling for Mrs Brooks to quit, and last week David Cameron said he would have accepted her resignation if she stood down.
In reaction to her leaving her job, the Prime Minister thought her resignation was "the right decision", his official spokesman has said.
Mrs Brooks announced she was going in an internal email to staff at the company, which also publishes The Times and Sunday Times.
In response James Murdoch said: "I want to thank her for her 22 years of service to the company.
"She has been one of the outstanding editors of her generation and she can be proud of many accomplishments as an executive. We support her as she takes this step to clear her name."
Chairman of the Commons Culture Committee John Whittingdale said he believed her resignation had been "inevitable".
"I think this is the right decision. I think many people expected it to come rather sooner, but I think her position was extremely difficult," he told Sky News.
Another committee member, Labour MP Tom Watson, who has been a leading critic over the phone-hacking scandal, said the spotlight would now shift to James Murdoch.
He told Sky News: "Her departure will now move the focus onto James Murdoch and what he knew about authorising payments to buy the silence of hacking victims."
Tom Mockridge, the chief executive of Sky Italia, has been named as her successor.
James Murdoch praised him as "a highly respected and accomplished media executive", who had shown "leadership and integrity" in creating the 24-hour TV news channel in Italy.