Texas obtained minor league pitching prospect Matt Ball in the deal announced Thursday. The right-hander will report to Class A Hickory.
Chicago is sending Ranaudo to Triple-A Charlotte along with right-hander Scott Carroll. The White Sox will make a roster move to fill Carroll's spot before Friday night's road game against the New York Yankees.
Ranaudo has allowed seven runs over two relief appearances this season. He gave up five in Tuesday's outing against the White Sox. Texas rallied for a 13-11 victory.
Ranaudo, 26, is 5-4 with a 6.33 ERA in 13 career appearances, including nine starts, with Boston and Texas.
Ball, 21, had 24 strikeouts in 16 innings in 10 relief outings at Class A Kannapolis. He was an 11th-round pick by the White Sox in 2013.