The 31-year-old had already announced his intention to leave the Scottish capital club following a 10-year stay.
Blair, who will win his 82nd cap against Fiji on Saturday, is joining a club that were relegated last season but are former winners of the Heineken Cup.
In a statement released by Edinburgh, Blair said: "I've never made a secret of my desire to play in France at some stage during my career and I'm delighted that it will be with Brive.
"It's a lovely place with an incredible support and I'm really looking forward to meeting up with them after the Scotland tour.
"There were a number of considerations that I wanted to take into account for me and my family and Brive ticks a lot of the boxes that matter to us."
Another player departing Edinburgh is Jim Thompson, who has signed for London Scottish.
Scotland A and Scotland Sevens international Thompson said: "It's a great honour to be joining London Scottish. The club is on the up and going places, and I'm looking forward to being part of a successful season."
The 27-year-old joins former Edinburgh team-mate Phil Godman as one of five Scottish summer signings the Exiles have made.
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