In this paper, we derive the fundamental properties of 1SWASPJ011351.29+314909.7 (J0113+31), a metal-poor (-0.40 ± 0.04 dex), eclipsing binary in an eccentric orbit (~0.3) with an orbital period of ~14.277 d. Eclipsing M dwarfs that orbit solar-type stars (EBLMs), like J0113+31, have been identified from their light curves and follow-up spectroscopy in the course of the WASP transiting planet search. We present the analysis of the first binary of the EBLM sample for which masses, radii and temperatures of both components are derived, and thus, define here the methodology. The primary component with a mass of 0.945 ± 0.045 M⊙ has a large radius (1.378±0.058 R⊙) indicating that the system is quite old, ~9.5 Gyr. The M-dwarf secondary mass of 0.186 ± 0.010 M⊙ and radius of 0.209 ± 0.011 R⊙ are fully consistent with stellar evolutionary models. However, from the near-infrared secondary eclipse light curve, the M dwarf is found to have an effective temperature of 3922 ± 42 K, which is ~600 K hotter than predicted by theoretical models. We discuss different scenarios to explain this temperature discrepancy. The case of J0113+31 for which we can measure mass, radius, temperature, and metallicity highlights the importance of deriving mass, radius, and temperature as a function of metallicity for M dwarfs to better understand the lowest mass stars. The EBLM Project will define the relationship between mass, radius, temperature, and metallicityfor M dwarfs providing important empirical constraints at the bottom of the main sequence. Light curves and radial velocity curve are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/572/A50