  An upper limit on the total annihilation cross section of dark matter (DM)
has recently been derived from the observed atmospheric neutrino background. We
show that comparable bounds are obtained for DM masses around the TeV scale by
observations of the diffuse gamma-ray flux by EGRET, because electroweak
bremsstrahlung leads to non-negligible electromagnetic branching ratios, even
if DM particles only couple to neutrinos at tree level. A better mapping and
the partial resolution of the diffuse gamma-ray background into astrophysical
sources by the GLAST satellite will improve this bound in the near future.
