"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," Mr Brown said in a statement.
"This tragedy reinforces our commitment to confront terrorism. It strengthens our determination never to concede to the demands of terrorists, nor to pay ransoms.
"I want those who would use terror against British citizens to know beyond doubt that we and our allies will pursue them relentlessly, and that they will meet the justice they deserve."
The group had threatened to kill Mr Dyer if the UK Government did not release Jordanian extremist Abu Qatada, who has been in prison since 2005 and is awaiting deportation to Jordan. A multimillion-euro ransom is also thought to have been demanded.
Al-Qaeda in Islamic Maghreb (AQIM) "killed Dyer on May 31, finding that Britain is unresponsive and does not seem to care for its citizens", the US-based monitoring SITE group quoted an online statement as saying.
"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," the statement said.
Al-Qaeda had on May 21 extended a deadline on its threat to kill Mr Dyer, giving the British government more time to respond to its demand to release Muslim cleric Abu Qatada, once regarded as Osama bin Laden's right-hand man in Europe.
Qatada has been held in Britain since 2005. He denies belonging to the group.
Britain's highest court ruled in February he could be deported to Jordan despite fears he may be tortured there.
Mr Dyer is believed to be one of a group of four European tourists who were kidnapped in January on the border between Mali and Niger after attending a Tuareg cultural festival at Anderamboukane in eastern Mali, near Timbuktu.
The others were two Swiss tourists and a German woman.
Mr Brown sent his condolences to the family of Mr Dyer, and said they would be offered "every possible support".
He said UK officials had worked "strenuously to avoid this outcome", and he had regularly discussed the situation with the president of Mali.
AQIM has claimed responsibility for the kidnapping of two Canadian diplomats and four European tourists in the past five months. The two diplomats and two of the tourists were freed in Mali in April.
Last month, Algerian media reported AQIM was demanding 10 million euros (£8.65million) in exchange for the Briton and a Swiss national being held in the Sahara.
