The 29-year-old came through Les Girondins' academy but failed to make a first-team appearance before joining Sedan on an initial loan in 2007.
He has since spent time at Brest, Saint-Etienne and most notably Nice before moving to Spain in the summer.
However, having struggled for consistency at La Rosaleda, he has now returned to his hometown club on a three-and-a-half-year deal and trained with his new team-mates on Thursday.
Baysse says finally getting his chance to play for Bordeaux means everything and he is excited about getting on to the field at the Matmut Atlantique.
"Thank you Bordeaux for fulfilling a childhood dream," Baysse tweeted. "I'm happy and proud to be wearing my colours again. I'm looking forward to starting the adventure."