We identify SDSS J121010.1+334722.9 as an eclipsing post-common-envelope binary, with an orbital period of ?, containing a very cool, low-mass, DAZ white dwarf and a low-mass main-sequence star of spectral type M5. A model atmosphere analysis of the metal absorption lines detected in the blue part of the optical spectrum, along with the Galaxy Evolution Explorer near-ultraviolet flux, yields a white dwarf temperature of ? K and a metallicity value of log [Z/H] =-2.0 ± 0.3. The Na I λλ8183.27, 8194.81 absorption doublet is used to measure the radial velocity of the secondary star, ?, and Fe I absorption lines in the blue part of the spectrum provide the radial velocity of the white dwarf, ?, yielding a mass ratio of q = 0.379 ± 0.009. Light-curve model fitting, using the Markov chain Monte Carlo method, gives the inclination angle as i = (79?05-79?36) ± 0?15, and the stellar masses as ? and ?. Systematic uncertainties in the absolute calibration of the photometric data influence the determination of the stellar radii. The radius of the white dwarf is found to be ? and the volume-averaged radius of the tidally distorted secondary is ?. The white dwarf in SDSS J121010.1+334722.9 is a very strong He-core candidate.