We report on the discovery of K2-141 b (EPIC 246393474 b), an ultra-short-period super-Earth on a 6.7 h orbit transiting an active K7 V star based on data from K2 campaign 12. We confirmed the planet's existence and measured its mass with a series of follow-up observations: seeing-limited MuSCAT imaging, NESSI high-resolution speckle observations, and FIES and HARPS high-precision radial-velocity monitoring. K2-141 b has a mass of 5.31 ± 0.46 M⊕ and radius of 1.54-0.09+0.10 R⊕, yielding a mean density of 8.00-1.45+1.83 g cm-3 and suggesting a rocky-iron composition. Models indicate that iron cannot exceed 70% of the total mass. With an orbital period of only 6.7 h, K2-141 b is the shortest-period planet known to date with a precisely determined mass. Based on observations obtained with (a) the Nordic Optical Telescope (NOT), operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos (ORM) of the Instituto de Astrofisica de Canarias (IAC); (b) the 3.6m ESO telescope at La Silla Observatory under program ID 099.C-0491; (c) the Kepler space telescope in its extended mission K2.Tables of the light curve data and the radial velocities 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/612/A95