Chemical vapor deposition synthesis of graphene on polycrystalline copper substrates from methane is a promising technique for industrial production and application. However, the quality of grown graphene can be improved by using single-crystal copper (111). In this paper, we propose to synthesize graphene on epitaxial single-crystal Cu film deposited and recrystallized on a basal-plane sapphire substrate. The effect of film thickness, temperature, and time of annealing on the size of copper grains and their orientation is demonstrated. Under optimized conditions, the copper grains with the (111) orientation and a record size of several millimeters are obtained, and the single-crystal graphene is grown over their entire area. The high quality of synthesized graphene has been confirmed by Raman spectroscopy, scanning electron microscopy, and the sheet resistance measurements by the four point probe method.