Laterally localized electronic states are identified on a single layer of graphene on ruthenium by low temperature scanning tunneling spectroscopy (STS). The individual states are separated by 3 nm and comprise regions of about 90 carbon atoms. This constitutes a highly regular quantum dot-array with molecular precision. It is evidenced by quantum well resonances (QWRs) with energies that relate to the corrugation of the graphene layer. The dI/dV conductance spectra are modeled by a layer height dependent potential-well with a delta-function potential that describes the barrier for electron penetration into graphene. The resulting QWRs are strongest and lowest in energy on the isolated 'hill' regions with a diameter of 2 nm, where the graphene is decoupled from the surface.