We obtain a monolayer graphene on epitaxial silicon carbide on silicon substrates via solid source growth mediated by a thin Ni–Cu alloy. Raman spectroscopy consistently shows an I D/I G band ratio as low as ~0.2, indicating that the graphene obtained through this method is to-date the best quality monolayer grown on epitaxial silicon carbide films on silicon. We describe the key steps behind the graphene synthesis on the basis of extensive physical, chemical and morphological analyses. We conclude that (1) the oxidation, amorphisation and silicidation of the silicon carbide surface mediated by the Ni, (2) the liquid-phase epitaxial growth of graphene as well as (3) the self-limiting graphitization provided the molten Cu catalyst, are key characteristics of this novel synthesis method.