  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-days 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 sigma_e < 3 x 10^{-38} cm^2 at 90%
CL, while dark matter masses between 20 MeV and 1 GeV are bounded by sigma_e <
10^{-37} cm^2 at 90% CL. 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.
