High-resolution angle-resolved photoemission reveals a complex electronic structure at the interface between a highly ordered C60 monolayer and the Cu(111) surface. The Shockley-type surface state of the clean substrate is pushed to higher energies by the overlayer and strongly couples to intramolecular C60 phonons. A second interface state with a bandwidth spanning the entire HOMO-LUMO gap disperses with the periodicity of the C60 superstructure. Density functional theory calculations indicate that this state is due to strong hybridization of molecular orbitals and substrate electronic states at the C60/metal interface. This finding sheds light on the issue of self-doping in molecular layers on metal surfaces.