France has confirmed that an Algerian jihadist group linked to Islamic State (IS) militants has beheaded tourist Herve Gourdel, seized on Sunday.
Jund al-Khilafa killed Mr Gourdel, 55, after its deadline for France to halt air strikes on IS in Iraq ran out.
French President Francois Hollande condemned the killing as a “cruel and cowardly” act.
He said that French air strikes which began on IS targets in Iraq last week would continue.
Speaking at the UN general assembly, Mr Hollande said that Mr Gourdel’s abduction and decapitation was a barbaric act of terrorism which presented a problem not only for the region but also for the world.
He said the fight against terrorism should know no borders and that France was now in mourning.
“It is not weakness that should be the response to terrorism but force,” he said.
Jund al-Khilafa posted a video of Mr Gourdel being killed which was entitled “Message of blood for the French government”.
IS itself has beheaded three Western hostages since August: US journalists James Foley and Steven Sotloff, and British aid worker David Haines. Their deaths were all filmed and posted online.
The group has also threatened to kill Alan Henning, a taxi driver from the UK, who was seized while on an aid mission to Syria in December.
On Sunday, it warned it would target Americans and other Western citizens, “especially the spiteful and filthy French”.
Mr Gourdel worked as a mountain guide in the Mercantour national park north of Nice, his home town.
He had also been organising treks through the Atlas Mountains of Morocco for some 20 years, AFP news agency reports.
The mayor of Nice, Christian Estrosi, said it was difficult to contain the “deep sadness” he felt.
“Today a war was declared on France,” he said. “We’ve been turning a blind eye to what’s happening in our back yards. And this is where it has led us.”
The BBC’s Lucy Williamson in Paris says news of Mr Gourdel’s killing has hit France hard.