We have grown well-ordered graphene adlayers on the lattice-matched Co(0001) surface. Low-temperature scanning tunneling microscopy measurements demonstrate an on-top registry of the carbon atoms with respect to the Co(0001) surface. The tunneling conductance spectrum shows that the electronic structure is substantially altered from that of isolated graphene, implying a strong coupling between graphene and cobalt states. Calculations using density functional theory confirm that structures with on-top registry have the lowest energy and provide clear evidence for strong electronic coupling between the graphene pi-states and Co d-states at the interface.