The equilibrium geometry and electronic structure of graphene deposited on a multilayer hexagonal boron nitride (h-BN) substrate has been investigated using the density functional and pseudopotential theories. We found that the energy band gap for the interface between a monolayer graphene (MLG) and a monolayer BN (MLBN) lies between 47 and 62 meV, depending on the relative orientations of the layers. In the most energetically stable configuration the binding energy is found to be approximately 40 meV per C atom. Slightly away from the Dirac point, the dispersion curve is linear, with the electron speed almost identical to that for isolated graphene. The dispersion relation becomes reasonably quadratic for the interface between MLG and 4-layer-BN, with a relative effective mass of 0.0047. While the MLG/MLBN superlattice is metallic, the thinnest armchair nanoribbon of MLG/MLBN interface is semiconducting with a gap of 1.84 eV.