The growth of graphene on single crystal Cu(111) has been achieved by thermal decomposition of ethylene in an ultrahigh vacuum chamber for the first time. The structural and electronic properties of graphene on Cu(111) have been investigated by scanning tunneling microscopy and spectroscopy. The nucleation of monolayer islands and two predominant domain orientations have been observed, which lead to the formation of numerous domain boundaries with increasing coverage. These results reveal that reducing the density of domain boundaries is one challenge of growing high-quality graphene on copper.