We combine models of the dynamical evolution of compact groups with models of the evolution of the stellar populations in merger products to predict the morphology and colors of galaxies in compact groups. Models in which a compact group forms out of a loose group and evolves by merging until becoming a single elliptical galaxy in several Gyr conflict with several observations. These models predict that about 15% of the elliptical galaxies in compact groups should be more than 0.10 mag bluer in (B-V)0 than normal for elliptical galaxies of their luminosity, whereas the observed value is lower (5.5(-3.0)+5.3%). In addition, 21 +/- 3% of the galaxies in compact groups are elliptical galaxies, but these models predict that only about 13% of the galaxies in compact groups should be elliptical galaxies. The models agree much better with the observations if compact groups form in environments with a greater fraction of early-type galaxies relative to normal loose groups. Additional constraints from the number of ongoing mergers in compact groups and the fraction of both normal and blue elliptical galaxies outside of compact groups require that the average time scale for the evolution of a compact group into a single elliptical galaxy is at least 4 Gyr.