There is a great deal of fundamental and practical interest in the possibility of inducing superconductivity in a monolayer of graphene. But while bulk graphite can be made to superconduct when certain metal atoms are intercalated between its graphene sheets, the same has not been achieved in a single layer. Moreover, there is a considerable debate about the precise mechanism of superconductivity in intercalated graphite. Here we report angle-resolved photoelectron spectroscopy measurements of the superconducting graphite intercalation compound CaC6 that distinctly resolve both its intercalant-derived interlayer band and its graphene-derived π* band. Our results indicate the opening of a superconducting gap in the π* band and reveal a substantial contribution to the total electron-phonon-coupling strength from the π*-interlayer interband interaction. Combined with theoretical predictions, these results provide a complete account for the superconducting mechanism in graphite intercalation compounds and lend support to the idea of realizing superconducting graphene by creating an adatom superlattice.