Electrically driven acousto-optic devices that provide beam deflection and optical frequency shifting have broad applications from pulse synthesis to heterodyne detection. Commercially available acousto-optic modulators are based on bulk materials and consume Watts of radio frequency power. Here, we demonstrate an integrated 3-GHz acousto-optic frequency shifter on thin-film lithium niobate, featuring a carrier suppression over 30 dB. Further, we demonstrate a gigahertz-spaced optical frequency comb featuring more than 200 lines over a 0.6-THz optical bandwidth by recirculating the light in an active frequency shifting loop. Our integrated acousto-optic platform leads to the development of on-chip optical routing, isolation, and microwave signal processing.