The Schr\"odinger equation dictates that the propagation of nearly free electrons through a weak periodic potential results in the opening of band gaps near points of the reciprocal lattice known as Brillouin zone boundaries. However, in the case of massless Dirac fermions, it has been predicted that the chirality of the charge carriers prevents the opening of a band gap and instead new Dirac points appear in the electronic structure of the material. Graphene on hexagonal boron nitride (hBN) exhibits a rotation dependent Moir\'e pattern. In this letter, we show experimentally and theoretically that this Moir\'e pattern acts as a weak periodic potential and thereby leads to the emergence of a new set of Dirac points at an energy determined by its wavelength. The new massless Dirac fermions generated at these superlattice Dirac points are characterized by a significantly reduced Fermi velocity. The local density of states near these Dirac cones exhibits hexagonal modulations indicating an anisotropic Fermi velocity.