  Motivated by a recent detection of 511 keV photons from the center of our
Galaxy, we calculate the spectrum of the soft gamma-ray background of the
redshifted 511 keV photons from cosmological halos. Annihilation of dark matter
particles into electron-positron pairs makes a substantial contribution to the
gamma-ray background. Mass of such dark matter particles must be <~ 100 MeV so
that resulting electron-positron pairs are on-relativistic. On the other hand,
we show that in order for the annihilation not to exceed the observed
background, the dark matter mass needs to be >~ 20 MeV. We include the
contribution from the active galactic nuclei and supernovae. The halo
substructures may increase the lower bound to >~ 60 MeV.
