David Cameron has vowed to do everything to defeat Islamic State (Isis) after the "absolutely appalling" murder of the British aid worker Alan Henning.
After talks with the security services and Foreign Office officials on Saturday at his country residence, the prime minister said the beheading made it clear there was "no level of depravity" to which Isis would not sink.
"It is senseless. It is complete unforgivable. Anyone in any doubt about this organisation can now see how truly repulsive it is and barbaric it is.
"This is going to be our struggle now, that with others we must do everything we can to defeat this organisation. We must take action against it and we must find those responsible."
Cameron called Henning, whose murder was depicted on a video Isis released on Friday night, a man of great peace, kindness and gentleness.
"He went with many Muslim friends out to do no more than simply help other people. His Muslim friends will be mourning him at this special time of Eid and the whole country is mourning with them."
The prime minister's comments were echoed by Nick Clegg, his deputy and the Liberal Democrat leader, who said Isis was "twisting ... religion to its own end".
"If you are doing this to intimidate us, you won't. If you are doing this to break our resolve, you won't. If you are doing this to drag us into a religious war on your terms, you won't," he said.
"What we in Britain will do instead in a calm and determined way is to play our role to hunt you down and to rid the world of this barbaric violence. What Isil is doing is sick and twisted and they will not win."
The US ambassador to Britain, Matthew Barzun, said on Saturday: "Today, we stand with the UK in joint determination, resolve and commitment to achieve justice for Alan Henning."
Barack Obama earlier condemned Henning's murder and said that Washington, along with its allies, would continue to take action to destroy Isis. "Henning worked to help improve the lives of the Syrian people and his death is a great loss for them, for his family and the people of the United Kingdom," the US president said on Friday night.
"Standing together with our UK friends and allies, we will work to bring the perpetrators of Alan's murder, as well as the murders of Jim Foley, Steven Sotloff and David Haines, to justice. Standing together with a broad coalition of allies and partners, we will continue taking decisive action to degrade and ultimately destroy Isil."
The former head of the British army said the murder of hostages such as Henning was "part of the harsh price" to be paid in confronting Isis.
Lord Dannatt, who stepped down in 2009, told BBC Radio 4's Today programme on Saturday:
"Our government is now standing up to its responsibilities, and part of the harsh price, while this takes time, is that sadly if they have hostages in their hands, these atrocious things seem to be happening and seem to be going to continue to happen."
Henning is the fourth western hostage Isis has killed, following the filmed beheadings of the US journalists James Foley and Steven Sotloff, and the Scottish aid worker David Haines.
A further hostage, Hervé Gourdel, from Nice in France, was murdered by Jund al-Khalifa, a group with links to Isis, on 24 September.
Dannat renewed his call for Britain to extend military operations to Syria, where the US and other allies are conducting air strikes. He said the UK was not doing enough.
"Dealing with half a problem is not going to solve the problem. If the Americans have found a way to be able to do this then actually we should be doing it," he said.
Henning's murder came after the UK launched air strikes against Isis, joining the US and its Arab allies - Jordan, Saudi Arabia and the UAE - which had been targeting the group for several days.
Shortly before the Ministry of Defence announced that the RAF had attacked two Isis targets, the Foreign Office arranged for Henning's wife, Barbara, to make a televised appeal for his release.
Intelligence agencies on both sides of the Atlantic are studying the video of his killing, which was released on the eve of the Islamic festival of Eid al-Adha.
On Friday night, the Foreign Office said: "We are aware of the video and are working urgently to verify the contents. If true, this is a further disgusting murder. We are offering the family every support possible; they ask to be left alone at this time."
Majid Freeman, a Leicester aid volunteer who worked with Henning in Syria, said he appeared to have been abandoned by the government.
Freeman feared that Britain's decision to join the US-led air strikes may have sealed Henning's fate. "Turkey managed to release 49 hostages without paying a single penny. All the British government had to do was release one hostage."
The time between the initial threat to kill Henning and his murder, longer than in previous cases, suggested that Isis had at least been open to sparing him, Freeman told the Today programme. "However the government have done nothing at all to help his case. If anything, they voted for air strikes which may have sealed his fate."
In the video, entitled Another Message to America and Its Allies, a masked man stands behind Henning, who is kneeling. Henning says: "I am Alan Henning. Because of our parliament's decision to attack the Islamic State, I - as a member of the British public - will now pay the price for that decision."
A voice then says: "The blood of David Haines was on your hands, Cameron. Alan Henning will also be slaughtered, but his blood is on the hands of the British parliament."
It ends with a threat to a fifth hostage, an US citizen named by his captors as Peter Edward Kassig.
Henning, 47, a taxi driver from Eccles, Greater Manchester, had been held captive in Syria for nine months, and is thought to have been held by Isis with up to 20 other western hostages for much of that time.