© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. We demonstrate sympathetic sideband cooling of a 40CaH+ molecular ion co-trapped with a 40Ca+ atomic ion in a linear Paul trap. Both axial modes of the two-ion chain are simultaneously cooled to near the ground state of motion. The center of mass mode is cooled to an average quanta of harmonic motion ncom = 0.13 ± 0.03 corresponding to a temperature of 12.47 ± 0.03 μk The breathing mode is cooled to nBM = 0.05 ± 0.02, corresponding to a temperature of 15.36 ± 0.01 μK.