The first direct detection limits on dark matter in the MeV to GeV mass range are presented, using XENON10 data. Such light dark matter can scatter with electrons, causing ionization of atoms in a detector target material and leading to single- or few-electron events. We use 15 kg day of data acquired in 2006 to set limits on the dark-matter-electron scattering cross section. The strongest bound is obtained at 100 MeV where σ(e)<3×10(-38) cm2 at 90% C.L., while dark-matter masses between 20 MeV and 1 GeV are bounded by σ(e)<10(-37) cm2 at 90% C.L. This analysis provides a first proof of principle that direct detection experiments can be sensitive to dark-matter candidates with masses well below the GeV scale.