Gant, 25, has a 3.83 ERA with 99 strikeouts in 103 1/3 innings over 18 starts for Memphis this season. His lone appearance with the big league club this year came on June 6, when he allowed the Reds two runs on three hits over 3 1/3 innings.
Siegrist was in the midst of a tough season with a 4.98 ERA over 34 1/3 innings and a 13.3 percent walk rate that ranked 192nd among 203 relievers with at least 30 innings. He had been battling left forearm tendinitis as well.
Siegrist, who was under team control for two more seasons, had fallen out of favor as a top left-hander in the Cardinals' bullpen with Brett Cecil, Tyler Lyons and Zach Duke also on deck.