We analyze the particle spectrum of the compact U(1) lattice gauge theory with a scalar matter field of unit charge. The nonperturbative Monte Carlo calculation is performed on lattices of sizes 83 × 16, 124 and 164. In the Coulomb phase we find a massless photon and massive scalar and vector bosonium states which are neutral bound states of charged bosonic particles, analogous to the positronium in QED. In the Higgs region of the confinement-Higgs phase the massive photon and the Higgs boson are present. Here we do not find any other vector state with a mass substantially different from the photon mass.