The second video, documenting the beheading, began circulating on Wednesday, according to a link provided by SITE Intelligence, which tracks jihadist groups.

France has a history of paying ransom through intermediaries to obtain the release of its kidnapped citizens, including four French journalists held by the Islamic State in Syria this year. However, the French government said this week that it would not give in to the Algerian cell’s ultimatum. The prime minister, Manuel Valls, told Europe 1 radio on Tuesday: “If we cede, if we retreat one inch, that would hand victory” to the militants.

Over the past decade, groups directly affiliated with Al Qaeda have collected millions of dollars in ransom from European governments and their proxies. According to the Treasury Department, about $40 million was paid last year to free four French citizens who were being held by Al Qaeda’s North African branch. An investigation by The New York Times found that Al Qaeda’s affiliates have generally moved away from killing their hostages for propaganda purposes, and prefer to use them as a source of revenue.

Image A recent photo of Mr. Gourdel in France. He was visiting Algeria for a hiking trip in the northern mountains. Credit... via Agence France-Presse — Getty Images

The exception to this trend was the Qaeda branch in Iraq, whose leader, Abu Musab al-Zarqawi, appeared to relish filming himself beheading his Western victims. His methods were deemed so brutal that the Qaeda leadership reprimanded him in a letter that was later made public, saying that he should kill his captives by shooting them, to avoid the horrifying and potentially damaging imagery of a decapitation.

Mr. Zarqawi’s group has since broken with Al Qaeda, and the Islamic State is its newest incarnation.

In the video circulated on Wednesday, Mr. Gourdel is shown kneeling before his four armed captors with his hands bound behind his back. His bony shoulders poke forward, and for the most part he stares at the ground as his kidnappers read a script announcing that he will be killed.