The Internet piracy enforcement agency Hadopi was mandated by the government in 2009 to send warning letters to ISP subscribers found to be downloading content without authorisation.
The law forces ISPs to hand over personal details of offending IP addresses after sifting through them to hunt out those breaching piracy regulations.
Hadopi was authorised to take legal action if a third warning was ignored.
It has so far sued only 14 people since the law came into force in October 2010.
A police court in the northeastern French city of Belfort fined the man for downloading songs of the Barbadian pop star Rihanna despite several warnings.
He had risked a 1,500-euro fine.
The man claimed it was his wife who had downloaded the music and he was fined for "negligence".
French President Francois Hollande has said he would like to replace Hadopi, an initiative of his predecessor Nicolas Sarkozy, with "something else".