  Loeb and Waxman have argued that high energy neutrinos from the decay of
pions produced in interactions of cosmic rays with interstellar gas in
starburst galaxies would be produced with a large enough flux to be observable.
Their model is reexamined here and we obtain an upper limit to the diffuse
neutrino flux from starburst galaxies. The upper limit obtained here is a
factor of ~5 lower than the flux which they predict. Our predicted neutrino
flux would be below the atmospheric neutrino foreground flux at energies below
\~300 TeV and therefore would be unobservable. PeV neutrinos from starburst
galaxies are also unlikely to be detected. Compared with predicted fluxes from
other extragalactic high energy neutrino sources, starburst neutrinos with ~PeV
energies would have a flux considerably below that predicted for AGN models. We
also estimate an upper limit for the diffuse GeV gamma-ray flux from starburst
galaxies to be about two orders of magnitude below the observed gamma-ray
background, much less than the background from blazars and more than an order
of magnitude below that calculated by Thompson et al.
