  The origin of the isotropic gamma-ray background (IGRB) --- the portion of
the extragalactic gamma-ray sky that is not resolvable into individual point
sources --- provides a powerful probe into the evolution of the high-energy
universe. Star-forming galaxies (SFGs) are among the most likely contributors
to the IGRB, though their contribution is difficult to constrain because their
flux distribution is dominated by numerous faint sources. We produce a novel
joint-likelihood analysis of the $\gamma$-ray emission from 584 SFGs, utilizing
advanced statistical techniques to compare the distribution of low-significance
excesses against the non-Poissonian $\gamma$-ray background fluctuations. We
first examine the theoretically well-motivated relationship between the far-IR
and $\gamma$-ray luminosities of SFGs, utilizing a model where the $\gamma$-ray
luminosity is given by log$_{10}$(L$_\gamma$/(erg s$^{-1}$)) = $\alpha$
log$_{10}$(L$_{IR}$/(10$^{10}$L$_\odot$)) + $\beta$. We calculate best-fit
parameters $\alpha$ = 1.18 $\pm$ 0.15, $\beta$ = 38.49 $\pm$ 0.24, with a
log-normal dispersion in this relationship given by $\sigma$ = 0.39 $\pm$ 0.12.
The best-fit values of $\alpha$ and $\beta$ are consistent with previous
studies. We find a larger dispersion in the far-IR to $\gamma$-ray correlation
than previous studies. This dispersion is significant at the level of
5.7$\sigma$. These results imply that SFGs significantly contribute to the
IGRB, producing between 61.0$^{+30.2}_{-18.3}\%$ of the total IGRB intensity
above an energy of 1 GeV. Along with recent works, this strongly indicates that
multiple source classes provide comparable contributions to the IGRB intensity.
We discuss the implication of these results for the interpretation of the
IceCube neutrinos.
