The case of Adelkhah and her French colleague and partner Roland Marchal, who were arrested together in June last year, has been a thorn in relations between Tehran and Paris for months.
Marchal was released in an apparent prisoner swap in March that drew strong criticism from the United States.
The 61-year-old Adelkhah has remained in custody ever since her arrest.
A research director at Sciences Po university in Paris, she is a dual French-Iranian citizen, a status Iran does not recognise. The academic was "sentenced to five years for gathering and conspiring against national security, and one year for propaganda against the Islamic republic," her lawyer Said Dehghan told AFP.
The sentences were to be served concurrently, he said, adding that his client intended to appeal against her conviction.
The punishment caused dismay in France.
"I strongly condemn (this verdict). This sentence is not based on any serious element... it has a political nature," Le Drian said in a statement.
Iranian authorities must "immediately free" Adelkhah and grant her a visit by French consular officials, he added.
Her trial opened on March 3 with the final hearing held on April 19 at branch 15 of Tehran's Revolutionary Court.
Adelkhah has been severely weakened by a 49-day hunger strike she mounted between late December and February, her lawyer said.