  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 orbiting solar-type stars (EBLMs), like J0113+31, have
been identified from WASP light curves and follow-up spectroscopy in the course
of the transiting planet search. We present the first binary of the EBLM sample
to be fully analysed, and thus, define here the methodology. The primary
component with a mass of 0.945 +/- 0.045 Msun has a large radius (1.378 +/-
0.058 Rsun) indicating that the system is quite old, ~9.5 Gyr. The M-dwarf
secondary mass of 0.186 +/- 0.010 Msun and radius of 0.209 +/- 0.011 Rsun 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 metallicity for M dwarfs
providing important empirical constraints at the bottom of the main sequence.
