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Based on photometric observations by MEarth and <jats:italic>Spitzer<\/jats:italic> as well as Doppler spectroscopy from the High Accuracy Radial velocity Planet Searcher, we report the discovery of an additional transiting rocky companion (LHS 1140 c) with a mass of 1.81\u00a0\u00b1\u00a00.39 <jats:italic>M<\/jats:italic>\n                  <jats:sub>\u2295<\/jats:sub> and a radius of 1.282\u00a0\u00b1\u00a00.024 <jats:italic>R<\/jats:italic>\n                  <jats:sub>\u2295<\/jats:sub> on a tighter, 3.77795-day orbit. We also obtain more precise estimates for the mass and radius of LHS 1140 b, which are 6.98\u00a0\u00b1\u00a00.89 <jats:italic>M<\/jats:italic>\n                  <jats:sub>\u2295<\/jats:sub> and 1.727\u00a0\u00b1\u00a00.032 <jats:italic>R<\/jats:italic>\n                  <jats:sub>\u2295<\/jats:sub>. The mean densities of planets b and c are 7.5\u00a0\u00b1\u00a01.0 g cm<jats:sup>\u22123<\/jats:sup> and 4.7\u00a0\u00b1\u00a01.1 g cm<jats:sup>\u22123<\/jats:sup>, respectively, both consistent with the Earth\u2019s ratio of iron to magnesium silicate. The orbital eccentricities of LHS 1140 b and c are consistent with circular orbits and constrained to be below 0.06 and 0.31, respectively, with 90% confidence. Because the orbits of the two planets are coplanar and because we know from previous analyses of <jats:italic>Kepler<\/jats:italic> data that compact systems of small planets orbiting M dwarfs are commonplace, a search for more transiting planets in the LHS 1140 system could be fruitful. 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