LHS 1140 is a nearby mid-M dwarf known to host a temperate rocky super-Earth (LHS 1140 b) on a 24.737-day orbit. Based on photometric observations by MEarth and Spitzer as well as Doppler spectroscopy from HARPS, we present the discovery of an additional transiting rocky companion (LHS 1140 c) with a mass of 1.81 ± 0.39 Earth masses and a radius of 1.282 ± 0.024 Earth radii on a tighter, 3.77795-day orbit. We also obtain more precise estimates of the mass and radius of LHS 1140 b to be 6.98 ± 0.98 Earth masses and 1.727 ± 0.032 Earth radii. The mean densities of planets b and c are 7.5 ± 1.0 g/cm3 and 4.7 ± 1.1 g/cm3, respectively, both consistent with the Earth's 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 co-planar and because we know from previous analyses of Kepler 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. LHS 1140 c is one of the few known nearby terrestrial planets whose atmosphere could be studied with the upcoming James Webb Space Telescope. This work was made possible with support from the David and Lucile Packard Foundation, the National Science Foundation, the John Templeton Foundation, NASA, and the Heising-Simons Foundation.