The 20-year-old has signed a five-year contract at the Stade Louis II, the Ligue 1 club have announced on their official website.
Monaco did not reveal the fee but reports in France say they have paid 20million euros to trigger the release clause in the player's contract.
Kondogbia spent just over a year in Andalusia, having moved from Lens at the start of last season, and scored once in 33 La Liga appearances for the club.
He was part of the France squad which won the Under-20 World Cup this summer for the first time and was handed a first call-up to Didier Deschamps' senior squad earlier this month against Belgium.
He was also included in the squad announced on Thursday for the upcoming World Cup qualifiers against Georgia and Belarus.