  The intrinsic distribution of spectral indices in GeV energies of
gamma-ray--loud blazars is a critical input in determining the spectral shape
of the unresolved blazar contribution to the diffuse extragalactic gamma-ray
background, as well as an important test of blazar emission theories. We
present a maximum-likelihood method of determining the intrinsic spectral index
distribution (ISID) of a population of gamma-ray emitters which accounts for
error in measurement of individual spectral indices, and we apply it to EGRET
blazars. We find that the most likely Gaussian ISID for EGRET blazars has a
mean of 2.27 and a standard deviation of 0.20. We additionally find some
indication that FSRQs and BL Lacs may have different ISIDs (with BL Lacs being
harder). We also test for spectral index hardening associated with blazar
variability for which we find no evidence. Finally, we produce simulated GLAST
spectral index datasets and perform the same analyses. With improved statistics
due to the much larger number of resolvable blazars, GLAST data will help us
determine the ISIDs with much improved accuracy. Should any difference exist
between the ISIDs of BL Lacs and FSRQs or between the ISIDs of blazars in the
quiescent and flaring states, GLAST data will be adequate to separate these
ISIDs at a significance better than 3 sigma.
