Red giant stars are perhaps the most important type of stars for Galactic and extra-galactic archaeology: they are luminous, occur in all stellar populations, and their surface temperatures allow precise abundance determinations for many different chemical elements. Yet, the full star formation and enrichment history of a galaxy can be traced directly only if two key observables can be determined for large stellar samples: age and chemical composition. While spectroscopy is a powerful method to analyse the detailed abundances of stars, stellar ages are the missing link in the chain, since they are not a direct observable. However, spectroscopy should be able to estimate stellar masses, which for red giants directly infer ages provided their chemical composition is known. Here we establish a new empirical relation between the shape of the hydrogen line in the observed spectra of red giants and stellar mass determined from asteroseismology. The relation allows determining stellar masses and ages with an accuracy of 10-15%. The method can be used with confidence for stars in the following range of stellar parameters: 4000<T_eff_<5000K, 0.5<logg<3.5, -2.0<[Fe/H]<0.3, and luminosities logL/L_{sun}_<2.5. Our analysis provides observational evidence that the H_{alpha}_ spectral characteristics of red giant stars are tightly correlated with their mass and therefore their age. We also show that the method samples well all stellar populations with ages above 1Gyr. Targeting bright giants, the method allows obtaining simultaneous age and chemical abundance information far deeper than would be possible with asteroseismology, extending the possible survey volume to remote regions of the Milky Way and even to neighbouring galaxies such as Andromeda or the Magellanic Clouds even with current instrumentation, such as the VLT and Keck facilities.