We show that the masses of red giant stars can be well predicted from their photospheric carbon and nitrogen abundances, in conjunction with their spectroscopic stellar labels log g, Teff, and [Fe/H]. This is qualitatively expected from mass-dependent post-main-sequence evolution. We here establish an empirical relation between these quantities by drawing on 1475 red giants with asteroseismic mass estimates from Kepler that also have spectroscopic labels from Apache Point Observatory Galactic Evolution Experiment (APOGEE) DR12. We assess the accuracy of our model, and find that it predicts stellar masses with fractional rms errors of about 14 per cent (typically 0.2 M⊙). From these masses, we derive ages with rms errors of 40 per cent. This empirical model allows us for the first time to make age determinations (in the range 1-13 Gyr) for vast numbers of giant stars across the Galaxy. We apply our model to ˜52 000 stars in APOGEE DR12, for which no direct mass and age information was previously available. We find that these estimates highlight the vertical age structure of the Milky Way disc, and that the relation of age with [α/M] and metallicity is broadly consistent with established expectations based on detailed studies of the solar neighbourhood.