The 28-year-old comes to Parkhead from Rennes looking to relaunch a career which, by his own admission, has stumbled in recent times. The former Auxerre and Paris St-Germain player has signed a three-year contract.
He has been heavily criticised for his recent performances, but believes that under Jansen's guidance, and with the support of the Parkhead faithful, he can show his qualities and rebuild his reputation.
Mahe, signed for a figure believed to be around pounds 500,000, becomes the first Frenchman to play for Celtic and, if he can put his personal problems behind him in Glasgow, should play a key role in improving the club's occasionally suspect back line.
Mahe said: "This is a big opportunity for me to re-launch my career and prove to my critics I can return to my best at a big club. I had a very happy time at Auxerre at the start of my career but at Paris St-Germain I had a few personal problems which held me back.''
Celtic general manager, Jock Brown, said: "Stephane can play in a traditional left-back role or as a modern wing-back depending on the tactics Wim Jansen wants to play.
"He is a player of considerable experience at a high level and we have tracked him for some time and are delighted to make him our fourth signing of recent weeks."
Fergus McCann, Celtic's managing director, added: "The board is pleased to support this continued process of restructuring of the side and delighted to obtain a quality player of international standard as Wim Jansen continues to reshape our squad in line with his plans."
Hibernian have made their second foreign signing in 24 hours with the transfer of experienced French defender Jean-Marc Adjovi-Boco from Lens. Jim Duffy, the Hibs manager, beat off competition from Premier Division rivals Motherwell and Kilmarnock to clinch Boco's signature on a two-year deal.
The move adds further strength to a Hibs defence already reinforced by yesterday's arrival of Icelandic international goalkeeper Olafur Gottskalksson from Keflavik. Both will go straight into the squad for Sunday's opening match against Celtic at Easter Road.
Boco, who can play as a full-back position or as a defensive midfielder, has played more than 200 games for Lens in six seasons of top-flight football in France.
He is now looking forward to the challenge of adjusting to the different demands of the Scottish League.
"The game here is much quicker and more physical than I am used to, but I think I can be a success in this sort of football," he said.Reuse content