Peter Moore was previously seen in a video released by the kidnappers
British hostage Peter Moore has been released alive after more than two-and-a-half years in captivity in Iraq.
Foreign Secretary David Miliband said the IT consultant from Lincoln, who was captured in Baghdad in May 2007, was in good health and "absolutely delighted".
His family said news of his release was "the best Christmas present ever" and "a big black cloud" had been lifted.
Four bodyguards were seized with Mr Moore. Three were shot dead; the fourth is also thought to have been killed.
The bodies of Jason Swindlehurst, from Skelmersdale, Lancashire, and Jason Creswell, of Glasgow, were returned to the UK in June 2009, followed by that of Alec MacLachlan, of Llanelli, Carmarthenshire, in September.
Fran and Pauline Sweeney said Mr Moore "sounded well and was cracking jokes"
Mr Miliband has called for the release of the body of the fourth guard - Alan McMenemy from Glasgow.
Mr Miliband said Mr Moore, who had been released on Wednesday morning, was in the British Embassy in Baghdad and would be reunited with his family as soon as possible.
Mr Moore's stepmother Pauline Sweeney told the BBC that hearing of his release was "the best Christmas present ever".
"We spoke to him this morning. I just burst into tears and kept asking if it was true. He sounds really well, very upbeat, he's cracking jokes," she said.
"He said when they came in to him this morning, he thought he was going out to a bullet in his head.
BBC security correspondent Frank Gardner
The sudden and dramatic release of Peter Moore has not come entirely out of the blue.
The Americans have only just handed over a key Shi'ite militant, Qais Al-Khaz'ali, to the Iraqi authorities after holding him since 2007.
This is the man the kidnappers always wanted back. They made it clear from the outset they wanted to trade prisoners: Qais and a handful of other Shi'ite militants for the five Britons.
British government policy has always been not to make substantive demands to kidnappers.
The tragedy for most of the British hostages and their families is that Iraq's political reconciliation process has only now reached a point where Qais could be released without appearing to give in to those kidnappers' demands.
"He had seen quite a few of the appeal broadcasts, which kept him going and made him realise there were people out there fighting for him and that he wasn't forgotten."
The foreign secretary said he had had a "very moving" conversation with Mr Moore, who was "to put it mildly, absolutely delighted". He is undergoing medical checks.
Mr Miliband said: "The joy and relief that will be felt by Peter's family will be mirrored by the continuing anguish of the family of Alan McMenemy.
"We have believed for some time that he has been killed and his family have been told our view of his likely fate."
The foreign secretary also told the BBC there were no concessions or deals made to secure Mr Moore's release.
"This was an Iraqi-led process of political reconciliation in which an armed group has made vows to come within the political system and to renounce violence, and that's the foundation of Peter Moore's release," he said.
David Miliband: "I am pleased to confirm that Peter Moore has been released today"
Mr Moore had been working for US management consultancy Bearingpoint in Iraq. The other men were security contractors employed to guard him.
The group was captured at the Iraqi Ministry of Finance by about 40 men disguised as Iraqi policemen.
They were understood to belong to an obscure militia known as the Islamic Shia Resistance, which demanded the release of up to nine of their associates held in US military custody since early 2007.
Several had already been handed to the Iraqi government and some had since been freed under the reconciliation process.
'Key' figure released
BBC security correspondent Frank Gardner said a senior Whitehall official had confirmed that Qais Al-Khaz'ali - the leader of the group that kidnapped the five Britons in Baghdad - was released "very recently" by the US to the Iraqi authorities.
"The main person the abductors wanted back, I'm told, is out of US custody... and handed over to the Iraqis," he said.
Qais Al-Khazaali had been suspected of involvement in the kidnapping and eventual killing of five US soldiers, Mr Gardner added.
Prime Minister Gordon Brown said: "I am hugely relieved by the wonderful news that Peter has been freed.
Alan McMenemy is believed to have been killed by his captors
"At this moment of celebration, we also remember the families of British hostages who have been killed in Iraq and elsewhere.
"And we pledge to continue to do everything we can to bring British hostages back to their loved ones."
Mr Moore's father Graeme, 60, from Wigston, Leicestershire, said he was "over the moon" at the news.
He said: "We are so relieved and we just want to get him home, back now to his family and friends.
"I'm breaking down, I'm just so overjoyed for the lad."
His natural mother, Avril Sweeney, 54, from Thornton Cleveleys, Lancashire, said: "It's like a big black cloud has been hanging over me and it has lifted now.
"I can smile again."
Terry Waite, who himself was a former hostage, said he was absolutely delighted at Mr Moore's release.
He added: "It is both a very happy time for Peter and his family and... while the other families will be delighted for him, there will be that tinge of sadness and one keeps them in mind at this point."
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