  In nuclei of starburst galaxies, the combination of an enhanced rate of
supernova explosions and a high gas density suggests that cosmic rays can be
efficiently produced, and that most of them lose their energy before escaping
these regions, resulting in a large flux of secondary products, including
neutrinos. Although the flux inferred from an individual starburst region is
expected to be well below the sensitivity of current neutrino telescopes, such
sources may provide a substantial contribution to the diffuse neutrino flux
measured by IceCube. Here we compute the gamma-ray and neutrino flux due to
starburst galaxies based on a physical model of cosmic ray transport in a
starburst nucleus, and accounting for the redshift evolution of the number
density of starburst sources as inferred from recent measurements of the star
formation rate. The model accounts for gamma-ray absorption both inside the
sources and in the intergalactic medium. The latter process is responsible for
electromagnetic cascades, which also contribute to the diffuse gamma-ray
background at lower energies. The conditions for acceleration of cosmic ray
protons up to energies exceeding $ \sim 10 \, \rm PeV$ in starburst regions,
necessary for the production of PeV neutrinos, are investigated in a critical
way. We show that starburst nuclei can account for the diffuse neutrino flux
above $\sim 200 \, \rm TeV$, thereby producing $\lesssim 40 \%$ of the
extragalactic diffuse gamma-ray background. Below $\sim 200 \, \rm TeV$, the
flux from starburst appears to be somewhat lower than the observed one, where
both the Galactic contribution and the flux of atmospheric neutrinos may
account for the difference.
