Relatives of murdered teenager Milly Dowler will "forgive but not forget" Rupert Murdoch after meeting with the media mogul.
Their family's solicitor Mark Lewis told journalists after the hastily arranged meeting that the 80-year-old media mogul had appeared "very sincere and shocked" when they met behind closed doors.
This afternoon Mr Murdoch met with the family of murdered teenager Milly Dowler, whose phone was allegedly hacked by a News of the World journalist.
"He was very humbled, very shaken and very sincere," the Dowler family's solicitor Mark Lewis said afterwards.
"This was something that had hit him on a very personal level, something that shouldn't have happened."
Messages on Milly's voicemail were allegedly deleted by News of the World journalists, leading her parents to briefly believe their daughter might still be alive.
"I don't think somebody could have held their head in their hands so many times to say they are sorry," Mr Lewis added.
The Dowlers have already met with prime minister David Cameron, deputy prime minister Nick Clegg and leader of the opposition Ed Miliband this week.
Mr Murdoch appeared in public for a few moments to confirm he had said sorry but refused to elaborate further, telling reporters the meeting had been "totally private". A protester shouting "shame on you" drowned out the bulk of his brief comments.
The Dowlers told Mr Murdoch that Mr Murdoch's papers should "lead the way to set the standard of honesty and decency in the field and not what had gone on before", Mr Lewis added.
"At the end of the day actions are going to speak louder than words."
Newspaper apologies this weekend
Mr Murdoch is using advertisements in national newspapers this weekend to offer a broader apology for the "hurt" caused by his company's journalists.
The News Corporation chief, who had faced criticism for not apologising before, is seeking to put the phone-hacking scandal behind him by finally expressing regret at the practices which took place under his leadership.
The text of the advertisement appearing in national newspapers over the weekend was released earlier today.
"We are sorry," the headline at the top of the advertisement reads.
"The News of the World was in the business of holding others to account. It failed when it came to itself.
"We are 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."
The last week has seen the media mogul shut down the News of the World tabloid, abandon his £10 billion bid to buy BSkyB and accept the resignation of News International chief executive Rebekah Brooks.
"I realise that simply apologising is not enough," Mr Murdoch adds.
"Our business was founded on the idea that a free and open press should be a positive force in society. We need to live up to this.
"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. Sincerely, Rupert Murdoch."