Adcock appeared in 36 games with the Royals from 2011-12, posting a record of 1-4 and a 3.79 ERA. The Rangers signed him to a Minor League contract in the offseason.
"He has been throwing the ball well," Rangers manager Ron Washington said. "He's a sinker/slider guy. He got hurt in Spring Training, but he is a multi-inning option for us."
To make room on the 40-man roster, the Rangers transferred first baseman Mitch Moreland to the 60-day disabled list. Moreland is out for the season after undergoing left ankle surgery.
T.R. Sullivan is a reporter for MLB.com. Read his blog, Postcards from Elysian Fields, and follow him on Twitter @Sullivan_Ranger. This story was not subject to the approval of Major League Baseball or its clubs.