The centre-back, who has made 130 appearances in Ligue 1, has been capped four times by France and was part of their squad at last year's World Cup.
"I am so happy to be here, at this big club," he said. "I am really proud to be able to be a part of this great family, and I hope to achieve very big things here, to win titles.
"I will do everything I can to achieve those objectives. I am very ambitious. I can't wait to feel the energy of the fans."
Co-sporting directors Laurence Stewart and Paul Winstanley said: "Axel has showcased his quality over several seasons in France and that has deservedly led to recognition on the international stage.
"He is ready to take the next step in his career and we are delighted that will be with Chelsea."
New Blues manager Mauricio Pochettino is overhauling his squad following the club's worst Premier League season in almost 30 years.
They open their new campaign at home to Liverpool on August 13th.
