Tony Blair welcomed the BBC's apology for errors in its Iraq weapons story, saying: "That's all I ever wanted".
Blair: Hopes line is drawn under the row
The prime minister said he wanted to make it "very clear" that he fully respected the independence of the BBC.
"I've no doubt the BBC will continue to probe and question the government in every proper way," he said.
"I think what this does now is it allows us to draw a line and move on - the BBC to get on with their job and the government to get on with ours."
Mr Blair was speaking in Hertfordshire, shortly after giving a speech on public service reform and within minutes of BBC director general Greg Dyke's resignation.
Earlier, BBC acting chairman Lord Ryder apologised "unreservedly" for any errors in the corporation's controversial Iraq weapons story which was a central focus of Lord Hutton's report.
Mr Blair said he welcomed Lord Ryder's statement on behalf of the BBC Board of Governors.
"This for me has always been a very simple matter. An accusation that was a very serious one ... was made. That was a false accusation, as Lord Hutton has found. It's now been withdrawn," he said.
"That's all I ever wanted."
He said he wanted to make it "absolutely clear" that he respected the BBC's independence and now hoped a line could be drawn under the whole affair.
Other reaction to Greg Dyke's resignation:
Lord Ryder of Wensum, Acting BBC Chairman:
"Gavyn Davies and Greg
Dyke served the public with strong and dynamic leadership. Both will be greatly
missed by the Board of Governors, the Executive Committee, the management and
staff with whom they worked.
"The manners of their departures demonstrates the integrity of both men. The
whole corporation owes them a debt of thanks and they deserve enormous credit
for their dedication and contributions to the BBC over the last few years."
Prime Minister's official spokesman:
Commenting on the resignations by both Mr Dyke and BBC chairman Gavyn Davies, he said: "The prime minister believes two decent and honourable men have done the decent and honourable thing and it is now time to move on."
Alex Salmond, parliamentary group leader for the Scottish National Party:
Called for a ministerial statement on the procedures for appointing a new chairman and director general to ensure the BBC's independence.
In the Commons, he said: "Will this honourable culture of resignation after mistakes
are made ... extend to the higher echelons of government if it transpires that
no weapons of mass destruction are found in Iraq?"
Ex-Tory Cabinet Minister Michael Portillo:
He was "very sorry" Mr Dyke had stepped down and added that he hoped the BBC would remain editorially independent.
"Greg is something of a friend of mine and I think he was very good for morale at the corporation.
"He's a very affable, jovial man and he obviously inspired people to work hard and he had the spirit of the BBC very much in mind and in his heart.
"I think I'm not surprised though - he was the editor in chief of the BBC and there was a great failure to check the story that Gilligan had put out."
John Stapleton, presenter GMTV:
Mr Dyke was a "very, very close friend" and he was sad for his colleagues at the BBC "who have lost a very good boss".
"Greg Dyke brings a breath of fresh air to any institution he walks into. What you see is what you get with Greg Dyke. There is nothing he loves more than being out on the newsroom floor with the guys talking about the stories.
"He is completely devoid of any of the pomposity that may have been a characteristic of some of his predecessors.
"I think the journalists there have lost a great friend, a man who acted in great faith on their behalf on what he though was good evidence at the time and it's just a terrible pity that he's gone."
Lord Rees Mogg, former BBC vice chairman:
Told BBC Five Live the corporation's apology was very full.
"The prime minister has accepted it. It seems to me that does bring this phase of the issue to an end.
"The Hutton Report, which I do not regard as satisfactory, was extremely severe. I think it would have been difficult for the director general to remain after that."
Ex-international development secretary Clare Short:
Told BBC News 24 she had a lot of respect for Greg Dyke but said his resignation had almost been inevitable.
"I think most people think the Hutton report was very, very unbalanced but now the most important issue is to protect the BBC," she said.
The BBC was an enormous national treasure which reached across the world and it was absolutely crucial that lessons were learned, she added.
Former MP and ex-BBC correspondent Martin Bell:
Told Five Live he thought Mr Dyke's resignation was unnecessary.
"I hope this cull is now going to end. There is no mileage in this for the government now.
"The BBC is a national treasure. OK it makes mistakes - it made a huge mistake - but I think there is a fundamental upswell of support for it now in the country.
"The government is going to do itself huge damage if it is seen to be attempting to intimidate the BBC any further."
Don Foster, the Liberal Democrat spokesman on culture, media and sport:
"Even after the Hutton report, 10 Downing Street has not let up. They've now had their pound of flesh with the resignations of Gavyn Davies and Greg Dyke and a profuse apology from the BBC.
"It is welcome that the prime minister has now agreed to draw a line underneath the affair but, like the BBC, 10 Downing Street should review its own arrangements.
"Alastair Campbell's accusation that 'large parts of the BBC' had an anti-war agenda, when independent research by Cardiff University showed BBC coverage was not biased, was as serious a charge as can be made against a public service broadcaster.
"The BBC can only remain independent if it is not suffocated by a climate of fear."
Labour MP Austin Mitchell:
Said it was a "disastrous" day for the BBC, for public service broadcasting, and for print journalism.
He said he thought the Hutton report was very bad, one sided and there was no reason for the BBC to apologise for everything.
"I've known many rows between the government and the BBC, but I've never seen one which has been conducted on the government's side with this lust to kill the BBC ... and producing such grovelling apologies from the BBC," he said.
Mr Mitchell said there was a danger that resignations opened up the doors to "everyone who hates the BBC ".
"Once you start giving unreserved apologies like this ... they're all in there for the kill. They've tasted blood and they want more," he said.
Stuart Purves, former editor in chief of ITN:
Told 5 Live it was regime change of a spectacular order.
"Only 24 hours ago we had the then chairman of the BBC raising a whole series of legitimate questions about the Hutton report ...
"Now we have a new acting chairman and a new acting director general - but the most concern I have is that statement that apologises unreservedly for errors."