Of all stoichiometric filled-skutterudite superconductors, LaRu4As12 has the highest critical field and temperature. Here we report on a detailed Fermi-surface investigation of LaRu4As12 by means of de Haas-van Alphen measurements and density-functional-theory calculations. We find evidence for a nearly spherical and a multiply connected Fermi-surface sheet. The different effective masses and mass enhancements for the two sheets support two-band superconductivity, which was inferred from previous specific-heat measurements. Furthermore, quantum oscillations persist as well in the superconducting phase. We use two models to describe the additional damping, yielding energy gaps differing by a factor of 5.