The Gunners captain undertook a medical before both clubs confirmed the move which sees him leave after nine years in north London.
"We thank Laurent for his contribution to the club and wish him all the best for the future," a statement read.
The club and Koscielny were thought to be in talks over a new contract but the defender was always keen to return to France.
He refused to go on the club's pre-season tour to the United States and put in a request to have his contract terminated to allow him to join another club, which was rejected by the Gunners.
Koscielny joined Arsenal in 2010 from Lorient and went on to make 353 appearances, and was awarded the captaincy in the 2016-17 season.
With the departure of Koscielny, Arsenal will be in the market for a defender and have been linked with Juventus' Daniele Rugani and RB Leipzig's Dayot Upamecano.
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