  We consider dark matter annihilation into Standard Model particles and show
that the least detectable final states, namely neutrinos, define an upper bound
on the total cross section. Calculating the cosmic diffuse neutrino signal, and
comparing it to the measured terrestrial atmospheric neutrino background, we
derive a strong and general bound. This can be evaded if the annihilation
products are dominantly new and truly invisible particles. Our bound is much
stronger than the unitarity bound at the most interesting masses, shows that
dark matter halos cannot be significantly modified by annihilations, and can be
improved by a factor of 10--100 with existing neutrino experiments.
