The family of Ken Bigley, the British man who was taken hostage in Iraq, says it has seen absolute proof that he has been killed.
Abu Dhabi television said it had been sent a video of Mr Bigley being beheaded but would not broadcast it.
Insurgent sources in Iraq say the 62-year-old was killed yesterday afternoon.
Earlier today his captors in a video said they were executing the hostage because the British government did not meet their demand to release Iraqi women detained in Iraq.
The Minister for Foreign Affairs, Dermot Ahern, said he believes the Irish Government did as much as it could to secure the release of Ken Bigley.
He said his sympathies at this time were with the Bigley family.
British Prime Minister Tony Blair expressed his 'utter revulsion' for the killing and declared that such actions 'in Iraq or elsewhere
should not prevail'.
The Taoiseach speaking from the Asia/Europe Summit in Vietnam also expressed his shock and outrage.
The British Foreign Secretary, Jack Straw, offered his condolences to the family after what he called the barbaric murder.
Mr Straw confirmed that the British Government had exchanged messages with Mr Bigley's captors through an intermediary, but that they would not abandon their demands.
The engineer was kidnapped in Baghdad on 16 September by the Tawhid and Jihad Group. Two Americans kidnapped by the same group were beheaded.
Earlier this week, the Government gave Mr Bigley an Irish passport in the hope it would persuade his captors to spare him.
