In a statement the group said it had executed the hostage late on Wednesday afternoon.
French armed forces had tried to rescue their agent at the weekend and it was believed he had been killed then. Two French commandos died in the failed raid.
"Let Muslims enjoy his execution and the French cry," the groups spokeman told one media outlet.
France has accused the militants of manipulating the media with their latest statements.
Al-Shabab, who had been holding Allex since July 2009, issued a statement saying he had been sentenced to death after what it called three years of "exhaustive attempts at negotiations" over his release.
Somalia has not had an effective central government for more than two decades.
France has a large military base in neighbouring Djibouti, including army, marine and air force units.
Al Shabab has been pushed out of all of the main towns they once controlled in southern and central parts of the country but they still remain a potent threat to the UN-backed government.
Source: Euronews
