Epitaxial graphene on Ir(111) prepared in excellent structural quality is investigated by angle-resolved photoelectron spectroscopy. It clearly displays a Dirac cone with the Dirac point shifted only slightly above the Fermi level. The moiré resulting from the overlaid graphene and Ir(111) surface lattices imposes a superperiodic potential giving rise to Dirac cone replicas and the opening of minigaps in the band structure.