Diplomats said they were told of Marie Dedieu’s death by contacts through whom they had been negotiating her release.
The exact date and circumstances of her death are not known, but her poor health and the fact medication had been withheld had led to fears this “tragic outcome was highly likely”, they said.
Mrs Dedieu, 66, was one of four Westerners recently taken from Kenya.
In September, Briton David Tebbutt was killed and his wife Judith abducted from a luxury resort of Kiwayu on the Kenyan coast.
Earlier this month, two female Spanish aid workers with the charity Medecins Sans Frontieres (MSF) were seized from the Dadaab refugee camp near the Kenya-Somalia border.
All three kidnapped women are still missing.
Mrs Dedieu, who was a wheelchair-user and had to take regular medication for cancer and heart problems, had lived part-time in Kenya since the 1990s.
She was taken from her beachfront home on the small island of Manda in the Lamu archipelago on 1 October by an armed gang.
Officials confirmed she had been taken by sea to Somalia and that the kidnappers did not take her wheelchair or medication with them.
In a statement, the French foreign ministry expressed its “indignation at the total lack of humanity and the cruelty shown by the kidnappers of our compatriot”.
It demanded that those responsible be brought to justice.
The Kenyan government said Mrs Dedieu died in the captivity of Somalia’s militant Islamist group, al-Shabab.
“The kidnapping and detention of Marie Dedieu was a terror act not only against her, but also against Kenya, her home country France and the entire world,” it said, in a statement.
But correspondents say it is not clear that al-Shabab, which controls most of south and central Somalia, abducted her.
Somalia has been wracked by fighting between various militias for two decades, so weapons are widely available and there are many armed groups who could be responsible.
Al-Shabab has not previously seized foreigners far from its own territory, while the numerous pirate gangs working out of Somalia normally kidnap ships and their crew for ransom rather than operating on land, correspondents say.
The abductors really are about as low as you can get. People should be free to live their lives in peace. For the gunmen, they had no political or religious cause to hide behind in seizing Marie Dedieu and the aid workers. All they are in it for is easy profit.
That part of Africa gets very bad press because of the actions of a few bandits. Yet all around these murdering scum there are hundreds of thousands of people earning an honest living and trying to make a better life for themselves. The bandits should follow this example and turn from terror as a source of income. If they don’t, newswarped sincerely hopes that the security forces who are defending the people manage to put a bullet in the head of as many bandits as they find in their sights.