We derive new spectral Hγ index definitions that are robust age indicators for old and relatively old stellar populations and therefore have great potential for solving the age-metallicity degeneracy of galaxy spectra. To study this feature as a function of age, metallicity, and resolution, we have used a new spectral synthesis model that predicts spectral energy distributions of single-age, single-metallicity stellar populations at resolution FWHM~1.8 Å (which can be smoothed to different resolutions), allowing direct measurements of the equivalent widths of particular absorption features. We show that Hγ's strong age-disentangling power is due to a compensating effect: at a specified age, Hγ strengthens with metallicity owing to an adjacent metallic absorption, but on the other hand, the adopted pseudocontinua are depressed by the effects of strong neighboring Fe I lines on both sides of Hγ. Despite the fact that this effect depends strongly on the adopted resolution and galaxy velocity dispersion σ, we propose a system of indicators that are completely insensitive to metallicity and stable against resolution, allowing the study of galaxies up to σ~300 km s-1. An extensive analysis of the characteristics of these indices indicates that observational spectra of very high signal-to-noise ratio and relatively high dispersion are required to gain this unprecedented age-discriminating power. Once such spectra are obtained, accurate and reliable estimates for the luminosity-weighted average stellar ages of these galaxies will become possible for the first time, without assessing their metallicities. We measured this index for two globular clusters, a number of low-luminosity elliptical galaxies, and a standard S0 galaxy. We find a large spread in the average stellar ages of a sample of low-luminosity ellipticals. In particular, these indices yield 4 Gyr for M32. This value is in excellent agreement with the age provided by the extraordinary fit to the full spectrum of this galaxy that we achieve in this paper.