We report a study of the valence band dispersion of twisted bilayer graphene using angle-resolved photoemission spectroscopy and ab initio calculations. We observe two noninteracting cones near the Dirac crossing energy and the emergence of van Hove singularities where the cones overlap for large twist angles (>5°). Besides the expected interaction between the Dirac cones, minigaps appeared at the Brillouin zone boundaries of the moiré superlattice formed by the misorientation of the two graphene layers. We attribute the emergence of these minigaps to a periodic potential induced by the moiré. These anticrossing features point to coupling between the two graphene sheets, mediated by moiré periodic potentials.