The Scotland international was out of contract after more than 100 appearances in almost 10 years at Warriors.
Dunbar moved to Newcastle Falcons on loan in January and joins Brive after they won France's second tier.
"He will bring all his experience at international level to evolve our young centres," said Brive head coach Jeremy Davidson.
Jamie Bhatti will be with Edinburgh next season but the prop forward says he didn't want to leave Scotstoun.
Speaking to the Scotland on Sunday, Bhatti said: "With the way I ended the season with Glasgow, obviously starting the (Guinness PRO14) semi and the final, I would love to have stayed on and played (there) but this is the way rugby is at times.
"When I was looking for a contract around Six Nations time, negotiations and stuff... ach they didn't want to sign me, they didn't want me at that point and they'd signed Oli (Kebble) and Alex (Allan) so then Edinburgh came in and I went and met with Richard (Cockerill) and had a coffee and really liked what he had to say."