British hostage Kenneth Bigley has been murdered by his captors in Iraq, his family has confirmed.
Kenneth Bigley was taken hostage on 16 September
A video released by militants showed Mr Bigley being killed, witnesses said.
"We can confirm that the family has now received absolute proof that Ken Bigley was executed by his captors," his brother Philip said.
Mr Bigley, 62, of Liverpool, was taken hostage on 16 September by the Tawhid and Jihad group. It beheaded two Americans captured at the same time.
Philip Bigley said the family had experienced "three long weeks of waiting and three long weeks of agony".
The government had done all it could to secure Mr Bigley's release, he said, but his brother's fate may have been sealed "from day one".
Foreign Secretary Jack Straw was visiting the Bigley family on Friday night.
He said the government had exchanged messages with the captors but they refused to abandon their demands.
Prime Minister Tony Blair expressed his "utter revulsion" at the killing.
The killing was also condemned by Iraq Prime Minister Iyad Allawi as a horrible, barbaric act.
There were also reports from the militant stronghold of Falluja that
Mr Bigley had been beheaded on Thursday afternoon in Latifiya, near Baghdad.
A video released by the kidnappers showed six hooded, armed men standing behind Mr Bigley, who was on his knees, according to a witness who watched the video for the Reuters news agency.
One of the six spoke in Arabic for a minute.
He said the group would carry out "the sentence of execution against this hostage" because the British government "did not meet our demand" to release Iraqi women detained by the US-led command in Iraq.
He then took a knife from his belt and cut off Mr Bigley's head as three other militants held him down.
The killer then held up the head for the camera.
Prime Minister Tony Blair said of the murder: "I feel desperately sorry for Ken Bigley, for his family who have behaved with extraordinary dignity and courage.
"I feel utter revulsion at the people who did this."
He said, the kidnappers had "played" the situation and their actions should not prevail over those "wanting to make Iraq a better place".
The Queen sent a private message of condolence to Ken Bigley's mother Lil this
evening, Buckingham Palace said.
Conservative leader Michael Howard condemned the kidnappers' "grotesque barbarity" adding: "We shall never give in to blackmail from terrorists."
Mr Bigley's relatives gathered at their family home in Liverpool as news of his death broke.
His son Craig, mother Lil and brothers Philip, Stan and Paul had campaigned for his release.
On Friday Philip Bigley thanked "those who have prayed for Ken and our family, from all religious backgrounds".
"We will always remember Ken for his love, compassion and, above all, his 'Liverpool' sense of humour," he said.
"He was a truly wonderful father, husband, brother and son. The loss to our family is immeasurable.
"The horror of these final days will haunt us for ever."
He said Mr Bigley's Liverpool-based family believed the government did everything possible to secure his release "in this impossible situation".
"It could be that the fate of Ken was sealed from day one," he said.
In a telephone interview with the TV station Al-Jazeera, Paul Bigley said he held the group who took his brother fully responsible for his death.
Mr Bigley said: "This war is illegal and it must stop. The United Nations declared this war illegal. There are no weapons of mass destruction, which were taken as a foundation for this war. Therefore, this war must stop."
Mr Bigley was last seen alive in a video broadcast last week, pleading for Tony Blair to help him.
Mr Bigley's Thai-born wife Sombat, had made several emotional appeals for mercy.
The engineer was captured just weeks before his official retirement when he would have joined her in Thailand.
Mr Bigley's Irish-born mother Lil, 86, has been treated in hospital several times.
Tawhid and Jihad is led by Jordanian militant Abu Musab al-Zarqawi.
The US says he has links to al-Qaeda and there is a £13.9m bounty on his head.
More than 30 of the 140 foreign nationals kidnapped in Iraq have been killed.