The masses of single molecular ions are nondestructively measured by cotrapping the ion of interest with a laser-cooled atomic ion, 40Ca+. Measurement of the resolved sidebands of a dipole forbidden transition on the atomic ion reveals the normal-mode frequencies of the two ion system. The mass of two molecular ions, 40CaH+ and 40Ca16O+, are then determined from the normal-mode frequencies. Isotopes of Ca+ are used to determine the effects of stray electric fields on the normal mode measurement. The future use of resolved sideband experiments for molecular spectroscopy is also discussed.