The widow of British aid worker Alan Henning said his family and friends were "numb with grief" after receiving the news that he had been murdered by his Islamic State captors.
Barbara Henning thanked everyone who took part in campaigns and vigils at home and abroad for the aid worker's release but appealed for privacy after receiving "the news we hoped we would never hear".
A video showing the brutal murder of the 47-year-old former taxi driver from Salford - who was kidnapped last December in Syria by IS militants - was posted on the internet by the group last night.
The beheading - apparently carried out by the same jihadist with an English accent behind those of three other Western captives including fellow British aid worker David Haines - drew wide international condemnation.
Prime Minister David Cameron vowed to use "all the assets we have" to hunt down and punish those behind the "senseless" act and insisted it would only reinforce the UK's determination to destroy IS.
Speaking after meeting with intelligence and defence chiefs, the Prime Minister said the killing of "a man of great peace, kindness and gentleness" showed that there was "no level of depravity to which they will not sink".
In a statement issued by the Foreign Office, Mrs Henning - who had issued emotional appeals for her husband's release - said: "Alan, my husband, and father of Lucy and Adam, was kidnapped in Syria in December last year.
"Last night we received news of his murder by Isil. It is the news we hoped we would never hear. As a family we are devastated by the news of his death. There are few words to describe how we feel at this moment. Myself, Lucy and Adam, and all of Alan's family and friends are numb with grief.
"On behalf of the entire family, I want to thank everyone who campaigned for Alan's release, who held vigils to pray for his safe return, and who condemned those who took him. Your efforts were a great support to us, and we take comfort in knowing how many people stood beside us in hoping for the best.
"Alan was a decent, caring human being. His interest was in the welfare of others. He will be remembered for this and we as a family are extremely proud of him and what he achieved and the people he helped.
"We now need time to come to terms with our loss. We would therefore be grateful if our privacy could be respected at this time."
Mrs Henning said support from the Foreign Office and from Greater Manchester Police "has been there from the start and has meant that we were able to get through the most awful of times".
"We always knew that Alan was in the most dangerous of situations but we hoped that he would return home to us. That is not to be."
Earlier, her brother Colin Livesey had accused the Government of doing too little to save his brother-in-law from the IS "scum" - who he said he prays "get what's coming to them".
"They could have done more when they knew about it months and months ago. Just the same with David Haines as well - I don't think they did enough for him either," he told BBC News.
Speaking of the militants behind the executions, h e said: "For a person who went there to give aid, to help their kids, their people in Syria, and for them to do what they have done, there is just no sense in it at all. I just hope and pray they get what's coming to them."
Mr Henning was killed despite a succession of appeals by both his family and influential Muslim figures at home and abroad, who warned the extremists that they would be acting in defiance of Islam if they killed him.
Mr Cameron said their failure to spare the life of an innocent man "of great peace, kindness and gentleness" showed there was "no level of depravity to which they will not sink".
"It is senseless, it's complete unforgivable," the PM said after the talks at his official country residence Chequers.
"Anyone in any doubt about this organisation can now see how truly repulsive it is and barbaric it is as an organisation.
"And as a country what we must do with our allies is everything we can to defeat this organisation in the region, but also to defeat it at home. And we must do everything we can to hunt down and find the people responsible for this."
In the video, another US hostage identified as Peter Kassig was paraded as the next intended victim.
Asked what action would be taken to track down the killers, amid calls for special forces to be sent in to find them, Mr Cameron said: "We will use all the assets we have, as we have been up to now, to try and find these hostages, to try and help these hostages, to help their families and do everything we can to defeat this organisation which is utterly ruthless, senseless and barbaric in the way it treats people."
In the video, IS sought to directly link the killing to the decision by the UK to join air strikes against the extremists. RAF Tornado jets have been involved in missions over the country for a week.
Mr Cameron said he believed Britain had the "resolve" to engage in a lengthy struggle to defeat IS - both through tackling extremism at home, removing videos such as that of Mr Henning's murder from the internet and taking part in the international military action.
"Everything we can do we will do but it's going to take, this struggle, patience, hard work and resolve to defeat this organisation. But I know this country, we have that resolve."
Number 10 said the talks centred on the "collective effort" with the US and regional allies to bring captors to justice " for their heinous crimes" and signalled that the murders would " not persuade us to change our approach".
"Indeed, the senseless murder of an innocent man only reinforces our resolve to defeat this terrorist organisation and to eradicate the threat they pose to Britons - whether those in the region or here on the streets of the UK."
The latest outrage will further increase pressure for the UK to extend its involvement beyond Iraq and join US-led air strikes on targets in Syria, something Mr Cameron held back from proposing last week because he was unsure of securing majority backing in the Commons.
Work and Pensions Secretary Iain Duncan Smith has said that cabinet ministers unanimously believe that it will eventually be necessary to take on the Islamic State terror group in its Syrian strongholds - blaming Labour reluctance for the limited action so far.
Labour leader Ed Miliband has called for efforts to secure United Nations Security Council backing for strikes within Syria.
But former head of the Army Lord Dannatt said there was no reason to hold back.
"Dealing with half a problem is not going to solve the problem," he told BBC Radio 4.
"If the Americans have found a way to be able to do this then actually we should be doing it."
Mr Miliband condemned the "appalling and barbaric" murder and said Labour would "do everything we can to support the efforts of the Government to bring those guilty of this terrible act to justice".
Deputy Prime Minister Nick Clegg said the Government would not be deflected from taking on IS.
"They will find some perverted pretext every time they wield a machete to murder innocent people in the future. None of the reasons they give, none, can ever justify what they do," he said.
The release of the footage came just hours after the father of British journalist John Cantlie, who is being held by IS, appealed for his son's release.
Paul Cantlie said in a video from his hospital bed: "He sought only to help the Syrian people and I ask you from all that is sacred to help us to allow him to return safely to those he loves and those who love him."
A second video released last night appearing to show a British IS fighter insulting Mr Cameron and challenging Western governments to send ground forces is also being investigated by the police, Number 10 said.
In it, he urges them to "send all your forces", claiming IS would "send them all back in coffins".
"The police are urgently investigating the contents of the video, including possible terrorism offences relating to it," a spokesman said.