The right-handed pitcher from Defiance, Ohio, was selected in the third round and signed a contract for $525,000.
Kime finished with a 6-1 record and 2.99 ERA for the Cardinals. He pitched 69.1 innings with 83 strikeouts in 27 appearances.
KIme was a junior and had one more season of eligibility.
Six other underclassmen – Jeff Thompson, Ty Young, Chad Green, Cody Ege, Coco Johnson and Adam Engel – were also drafted.