Image copyrightAFPImage captionFather Jacques Hamel was giving morning Mass when the attackers stormed his church
An 84-year-old priest was killed and four other people taken hostage by two armed men who stormed his church in a suburb of Rouen in northern France.
The two attackers, who said they were from the so-called Islamic State (IS), slit Fr Jacques Hamel's throat during a morning Mass, officials say.
Police later surrounded the church, in St-Etienne-du-Rouvray, and shot dead both hostage-takers.
One of the hostages is in a critical condition in hospital.
European terror attacks
President Francois Hollande, visiting the scene, said the attackers had committed a "cowardly assassination" and France would fight IS "by all means".
Pope Francis decried the "pain and horror of this absurd violence".
One attacker is reported to have been known to French police, and had tried to enter Syria last year.
Police special forces raided a house in Saint-Etienne-du-Rovray in the aftermath of the attack, and French prosecutors earlier said one person had been arrested.Image copyrightAPImage captionFive people were inside the church when it was attackedImage copyrightREUTERSImage captionSpecial police forces raided a house in the town after the attack
The attack happened during morning Mass at the historic church, situated in a quiet square of St-Etienne-du-Rouvray.
One nun who witnessed the attack, told BFM TV that the men forced Fr Hamel to his knees and recorded the killing, speaking Arabic as they did so. She says she fled as they were preparing to kill him.
Elite police units, specialised in hostage-taking, surrounded the church.
President Hollande said the attackers claimed to be from the self-styled IS before they were killed by police as they came out of the church.
Three of the hostages were freed unharmed, but one remains in a critical condition, said French interior ministry spokesman Pierre-Henri Brandet.
Within hours of the attack, the IS-linked Amaq news agency, said "two IS soldiers" had carried out the attack.
Few details are yet known about the attackers, but Mohammed Karabila, a local Muslim leader, told the Associated Press that one of them had been "followed by police for at least a year and a half".
The French ITele website also said one of them had tried to reach Syria in May 2015 but was turned back at the Turkish border.
According to the report, he then spent nearly a year in prison before being released in March, on condition he wear an electronic tag and move back in with his parents.
Analysis - BBC security correspondent Frank Gardner
After criticisms of police shortcomings over the Nice attack on 14 July, French anti-terrorist police moved with commendable speed to close down this hostage situation in a Norman church.
Observers have pointed out that France and other countries are now moving more quickly towards what is called tactical intervention - overwhelming armed force aimed at minimising the period during which terrorists or armed criminals can threaten the public.
The selection of a church by the attackers, whom IS refer to as its "soldiers", crosses a new red line in the grim history of recent attacks on continental Europe. The murder of a defenceless 84-year-old priest in this attack will have further inflamed public opinion.
News that one of the attackers was on the French government's terror watch-list, known as the S list, will prompt many to question its purpose if he can remain at large to carry a knife into a church.
Residents of St-Etienne-du-Rouvray reacted with shock and sadness to the killing of Fr Hamel, a well-known figure in the community.
"My family has lived here for 35 years and we have always known him," said Eulalie Garcia.
"He was someone who was treasured by the community. He was very discreet and didn't like to draw attention to himself."
Pensioner Claude-Albert Seguin told AP: "Everyone knew him very well. He was very loved in the community and a kind man."
French Prime Minister Manuel Valls has expressed his horror at the "barbaric attack" and said: "The whole of France and all Catholics are wounded. We will stand together."
The Archbishop of Rouen, Dominique Lebrun, who was attending a Catholic gathering in Poland, said: "I cry out to God with all men of goodwill. I would invite non-believers to join in the cry.
"The Catholic Church cannot take weapons other than those of prayer and brotherhood among men."
France is still reeling from the Bastille Day attack in Nice earlier this month, when a lorry was driven into celebrating crowds by Tunisian Mohamed Lahouaiej-Bouhlel, killing more than 80 people.
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