Graphene has shown great application potential as the hostmaterial for next-generation electronic devices. However, despite itsintriguing properties, one of the biggest hurdles for graphene to beuseful as an electronic material is the lack of an energy gap in itselectronic spectra. This, for example, prevents the use of graphene inmaking transistors. Although several proposals have been made to open agap in graphene's electronic spectra, they all require complexengineering of the graphene layer. Here, we show that when graphene isepitaxially grown on SiC substrate, a gap of ~;0.26 eV is produced. Thisgap decreases as the sample thickness increases and eventually approacheszero when the number of layers exceeds four. We propose that the originof this gap is the breaking of sublattice symmetry owing to thegraphene-substrate interaction. We believe that our results highlight apromising direction for band gap engineering of graphene.