We present the detection of a warm Neptune orbiting the M dwarf Gl 378, using radial velocity measurements obtained with the SOPHIE spectrograph at the Observatoire de Haute-Provence. The star was observed in the context of the SOPHIE exoplanet consortium’s sub-programme dedicated to finding planets around M dwarfs. Gl 378 is an M1 star, of solar metallicity, at a distance of 14.96 pc. The single planet detected, Gl 378 b, has a minimum mass of 13.02 M_Earth and an orbital period of 3.82 days, which place it at the lower boundary of the hot Neptune desert. As one of only a few such planets around M dwarfs, Gl 378 b provides important clues to the evolutionary history of these close-in planets. In particular, the eccentricity of 0.1 may point to a high-eccentricity migration. The planet may also have lost part of its envelope due to irradiation.