The Somali Islamist group al-Shabab says it has killed French intelligence agent Denis Allex in retaliation for a failed French operation to free him.
The French government has said it believes Mr Allex was killed during last week's raid, in which two French commandos also died.
Mr Allex - a French spy - was kidnapped in Somalia in July 2009.
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Al-Shabab, which is affiliated to al-Qaeda, said on its Twitter account that Mr Allex had been killed on Wednesday.
Al-Shabab had earlier threatened to kill Mr Allex and said that because of the rescue attempt, France would be responsible for his death.
The group has also cited France's intervention against Islamists in Mali in its decision to kill Mr Allex.
On Friday, about 50 French commandos launched an assault by helicopter on the al-Shabab stronghold of Bulo Marer, believing Mr Allex was being held in the town.
Al-Shabab said it had advance warning of the attack and Mr Allex - a codename for the French intelligence agent - was not in Bulo Marer at the time.
France says 17 militants were killed during the fighting, which witnesses said was intense and lasted for at least an hour.
Several civilians were reported to have been killed in the clashes.
Al-Shabab said it has taken the decision to execute their hostage to avenge the civilians killed during the French operation.
France said it launched the mission after repeated attempts to negotiate Mr Allex's release failed.
Somalia's government said it had no prior knowledge of the raid and it regretted the loss of civilian lives.
On Monday, al-Shabab published a photograph of a French soldier who the group said had died of gunshot wounds after being captured during the raid.
France has a large military base in neighbouring Djibouti, including army, marine and air force units.
For more than 20 years Somalia has seen clan-based warlords, rival politicians and Islamist militants battling for control of the country.
Last year, MPs elected a new president in a process backed by the UN.
Over the last 18 months, African Union forces working with Somali government troops have pushed the al-Shabab militants out of the major towns they controlled in central and southern Somalia.
But the group still controls a large area, including many smaller towns.