The deal was made Saturday.
Badenhop turns 30 on Feb. 8. He was 3-2 with a 3.03 ERA in 66 games last season. He has been a full-time reliever for the last three seasons and is 16-17 with two saves and a 4.08 ERA in his career with the Marlins and Rays.
The 20-year-old Mondesi, son of a former All-Star outfielder, hit .231 with five home runs and 32 RBIs in 68 games at rookie league Helena.