The properties of crystalline solids can to a large extent be derived from the scale and dimensionality of periodic arrays of coupled quantum systems such as atoms and molecules. Periodic quantum confinement in two dimensions has been elusive on surfaces, mainly because of the challenge to produce regular nanopatterned structures that can trap electronic states. We report that the two-dimensional free electron gas of the Cu(111) surface state can be trapped within the pores of an organic nanoporous network, which can be regarded as a regular array of quantum dots. Moreover, a shallow dispersive electronic band structure is formed, which is indicative of electronic coupling between neighboring pore states.