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 to determine stellar masses and ages with the accuracy of 10-15%. The method can be used with confidence for stars in the following range of stellar parameters: 4000 < Teff < 5000 K, 0.5 < log g < 3.5, -2.0 < [Fe/H] < 0.3, and luminosities log L/LSun < 2.5. Our analysis provides observational evidence that the Halpha 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 1 Gyr. Targeting bright giants, the method allows to obtain 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 like Andromeda or the Magellanic Clouds already with present instrumentation, like VLT and Keck facilities.