  Cosmic ray protons interacting with gas at the mean density of the
interstellar medium in starburst galaxies lose energy rapidly via inelastic
collisions with ambient nuclei. The resulting pions produce secondary electrons
and positrons, high-energy neutrinos, and gamma-ray photons. We estimate the
cumulative gamma-ray emission from starburst galaxies. We find a total
integrated background above 100 MeV of F_gamma ~ 10^{-6} GeV/cm^2/s/sr and a
corresponding specific intensity at GeV energies of nuI_nu ~ 10^{-7}
GeV/cm^2/s/sr. Starbursts may thus account for a significant fraction of the
extra-galactic $\gamma$-ray background. We show that the FIR-radio correlation
provides a strong constraint on the gamma-ray emission from starburst galaxies
because pions decay into both gamma-rays and radio-emitting electron/positron
pairs. We identify several nearby systems where the potential for observing
gamma-ray emission is the most favorable (M82, NGC 253, and IC 342), predict
their fluxes, and predict a linear FIR-gamma-ray correlation for the densest
starbursts. If established, the FIR-gamma-ray correlation would provide strong
evidence for the ``calorimeter'' theory of the FIR-radio correlation and would
imply that cosmic rays in starburst galaxies interact with gas at approximately
the mean density of the interstellar medium (ISM), thereby providing an
important constraint on the physics of the ISM in starbursts.
