On the basis of first-principles calculations, we report that a novel interfacial atomic structure occurs between graphene and the surface of silicon carbide, destroying the Dirac point of graphene and opening a substantial energy gap there. In the calculated atomic structures, a quasiperiodic 6x6 domain pattern emerges out of a larger commensurate 6 sqrt [3] x 6 sqrt [3]R30 degrees periodic interfacial reconstruction, resolving a long standing experimental controversy on the periodicity of the interfacial superstructures. Our theoretical energy spectrum shows a gap and midgap states at the Dirac point of graphene, which are in excellent agreement with the recently observed anomalous angle-resolved photoemission spectra. Beyond solving unexplained issues in epitaxial graphene, our atomistic study may provide a way to engineer the energy gaps of graphene on substrates.