White, 32, is slashing .233/.285/.406 with home runs, 17 RBI, and four stolen bases in 69 games this season, while Uberstine, 27, is tossing a 3.23 ERA through 39 innings with 48 strikeouts in Triple-A.
Uberstine, who also had one appearance for the Sox this season, was optioned to Triple-A by the Braves after the trade was confirmed.