We use scanning tunneling spectroscopy (STS) at low temperatures to investigate the local electronic structure of mono- and bilayer graphene grown epitaxially on SiC(0001). Already for monolayer graphene, a gap opening is observed in the pi-bands at the Dirac point. The gap size is spatially modulated with the (6 root 3 x 6 root 3)R30 degrees periodicity of the interface structure. We ascribe this effect to a spatially dependent interface potential, which is imprinted into the graphene layer. For bilayer graphene the Dirac gap has a constant size, but a spatially localized mid-gap state is observed within. For both, gap state and pi-bands the intensities are strongly modulated with the atomic periodicity of graphene. (C) 2008 Elsevier B.V. All rights reserved.