Militants released this photograph of Edwin Dyer in February
Downing Street says there is "strong reason to believe" that a British citizen has been killed by al-Qaeda militants in north-west Africa.
Edwin Dyer was kidnapped in Niger in January, but was being held in Mali.
The group had said it would kill Mr Dyer if the British government refused to release radical Muslim cleric Abu Qatada from a UK prison.
UK Prime Minister Gordon Brown condemned what he called an "appalling and barbaric act of terrorism".
He later told the Commons: "I want those who use terror against this country, and against British citizens, to know beyond doubt that they will be hunted down and brought to justice.
"There will be no hiding place for them and no safe haven for terrorists who attack our country."
Abu Qatada is awaiting extradition to Jordan, where he was convicted of terrorism offences in his absence and faces life in jail.
Mr Dyer, who spoke fluent German and had been working in Austria, was kidnapped in Niger on 22 January, close to the border with Mali.
He was captured along with a number of other European tourists, including two Swiss and one German. The group had been visiting the Anderamboukane festival on nomad culture.
In a statement issued on Wednesday, Mr Brown said: "We have strong reason to believe that a British citizen, Edwin Dyer, has been murdered by an al-Qaeda cell in Mali.
"I utterly condemn this appalling and barbaric act of terrorism.
"My thoughts are with Edwin Dyer's family. I offer them the condolences of the whole country."
Mr Brown said the killing reinforced Britain's "commitment to confront terrorism".
"It strengthens our determination never to concede to the demands of terrorists, nor to pay ransoms," he added.
Tory leader David Cameron backed the prime minister's stance.
He told MPs: "This must be a simply horrific time for his family and I'm sure everyone in the country is thinking about them," he said.
"In spite of all the difficulties, though, the prime minister is right; we must never give into terrorists."
Most of the other hostages have been released, but one Swiss man remains in captivity and the British government urged the kidnappers to release him "immediately and unconditionally".
BBC security correspondent Frank Gardner said Mr Dyer was believed to have been kidnapped by local tribesmen and sold on to Algerian members of al-Qaeda in Mali.
British officials had been using intermediaries to try to secure Mr Dyer's release, our correspondent said.
Other European countries are understood to have paid ransoms to secure their hostages' release, but the British government has a blanket policy against such methods, he added.
No further details about Mr Dyer have been released by the Foreign Office.
The militants posted a statement on an Islamist website announcing the killing.
"The British captive was killed so that he, and with him the British state, may taste a tiny portion of what innocent Muslims taste every day at the hands of the Crusader and Jewish coalition to the east and to the west," it said.
In a posting last month, the group said Abu Qatada must be released within 20 days or Mr Dyer would be killed.
The Foreign Office advises against all travel to parts of Mali and Niger. Its website states that there is "a high threat of kidnapping" and a "high threat from terrorism", especially in the border region.
Foreign Secretary David Miliband said: "Hostage-taking and murder can never be justified whatever the cause.
"This tragic news is despite the strenuous efforts of the UK team in the UK and Mali, with valuable help from international partners."
Abu Qatada is awaiting extradition to Jordan, where he faces life in jail
The Swiss Federal Department of Foreign Affairs said it condemned "this extreme violation of human dignity".
"Our thoughts are with the grieving relatives in Great Britain and the relatives of the Swiss hostage who continue to hope with all their hearts for the speedy release and return of their loved one," it said in a statement.
"No true religion and no world view dedicated to the well-being of all humans can condone the killing of innocent people."
Abu Qatada was once described as "Osama bin Laden's right-hand man in Europe".
He was granted asylum in the UK in 1994, but went on the run in 2001 on the eve of government moves to introduce new anti-terror laws allowing suspects to be detained without charge or trial.
In October 2002, he was caught and taken to Belmarsh Prison, but was freed on bail in March 2005, subject to a control order.
He was taken back into custody in August that year and held until June 2008. After another short period of freedom, he was detained again in December last year and remains in Long Lartin prison in Worcestershire, pending extradition.
Dr Muhammad Abdul Bari,, secretary general of the Muslim Council of Britain, said the "senseless murder" of Mr Dyer was "unforgivable".
"There can be no possible justification for this appalling action," he added.