The properties of photoemission electron sources determine the ultimate performance of a wide class of electron accelerators and photon detectors. To date, all high-efficiency visible-light photocathode materials are either polycrystalline or exhibit intrinsic surface disorder, both of which limit emitted electron beam brightness. In this Letter, we demonstrate the synthesis of epitaxial thin films of Cs_{3}Sb on 3C-SiC (001) using molecular-beam epitaxy. Films as thin as 4 nm have quantum efficiencies exceeding 2% at 532 nm. We also find that epitaxial films have an order of magnitude larger quantum efficiency at 650 nm than comparable polycrystalline films on Si. Additionally, these films permit angle-resolved photoemission spectroscopy measurements of the electronic structure, which are found to be in good agreement with theory. Epitaxial films open the door to dramatic brightness enhancements via increased efficiency near threshold, reduced surface disorder, and the possibility of engineering new photoemission functionality at the level of single atomic layers.