Dempster, who boasts the second-best ERA in the National League at 2.25, went to the AL West-leading Rangers in exchange for Class A third-baseman Christian Villaneuva and pitcher Kyle Hendricks.
Dempster, 35, originally drafted by the Rangers in 1995, has compiled a 5-5 record over 16 starts for the last-place Cubs.
He reportedly had demanded a trade to the Los Angeles Dodgers and was unwilling to waive his right to refuse a trade to anyone else, but ultimately agreed to join the defending AL Champion Rangers.
The right-hander, who was on the disabled list for three weeks with a strained lat muscle in June and July, has a 117-121 record with a 4.31 career ERA over a 15-year career with the Florida Marlins, Cincinnati Reds and Cubs.
2014: The Year in Music [PHOTOS]