  We report the discovery from K2 of a transiting terrestrial planet in an
ultra-short-period orbit around an M3-dwarf. K2-137 b completes an orbit in
only 4.3 hours, the second-shortest orbital period of any known planet, just 4
minutes longer than that of KOI 1843.03, which also orbits an M-dwarf. Using a
combination of archival images, AO imaging, RV measurements, and light curve
modelling, we show that no plausible eclipsing binary scenario can explain the
K2 light curve, and thus confirm the planetary nature of the system. The
planet, whose radius we determine to be 0.89 +/- 0.09 Earth radii, and which
must have a iron mass fraction greater than 0.45, orbits a star of mass 0.463
+/- 0.052 Msol and radius 0.442 +/- 0.044 Rsol.
