  The annihilation of new dark matter candidates with masses $m_X$ in the MeV
range may account for the galactic positrons that are required to explain the
511 keV $\gamma$-ray flux from the galactic bulge. We study the impact of
MeV-mass thermal relic particles on the primordial synthesis of $^2$H, $^4$He,
and $^7$Li. If the new particles are in thermal equilibrium with neutrinos
during the nucleosynthesis epoch they increase the helium mass fraction for
$m_X\alt 10$ MeV and are thus disfavored. If they couple primarily to the
electromagnetic plasma they can have the opposite effect of lowering both
helium and deuterium. For $m_X=4$--10 MeV they can even improve the overall
agreement between the predicted and observed $^2$H and $^4$He abundances.
