The carbon, nitrogen, and oxygen abundances of 22 massive, Galactic, A-type supergiants are presented in order to study the evolutionary status of these stars. Various evolution scenarios describe vastly different histories for these stars, which can be distinguished by the atmospheric abundances of CNO that they predict. Chemical abundances are determined from observations of weak optical spectral lines gathered at the McDonald Observatory. An atmospheric analysis is performed for each star adopting the most recent Kurucz LTE model atmospheres. Atmospheric parameters and LTE metal abundances for these stars have been presented in Venn (1995). Non-LTE (NLTE) line formation calculations have been carried out for the nitrogen and carbon abundances. For carbon, we adopt the Sturenburg & Holweger (1990) model atom and extensively test the effects of the atomic data on the resultant NLTE corrections. We find significant NLTE corrections [=log epsilon(X)(NLTE) - log epsilon(X)(LTE)] for the cooler supergiants that range from -0.1 in the F0 stars to -0.5 in the A3 stars. The mean NLTE carbon abundance is log epsilon(C)(NLTE) = 8.14 +/- 0.13 for the 14 A3-F0 supergiants. For the hotter stars, we show that the only C I lines that we have observed near 9100 Angstrom do not yield reliable elemental abundances. For nitrogen, we have constructed a new, detailed model atom (Lemke & Venn 1995). Application of this model atom to the A-type supergiants shows that departures from LTE strongly affect the nitrogen abundances. NLTE corrections for weak lines are quite large, ranging from -1.0 dex in the A0 supergiants to -0.3 in the FO supergiants. The average NLTE nitrogen abundance is log epsilon(N)(NLTE) = 8.05 +/- 0.19 for the 22 A0-F0 supergiants. These NLTE abundances do not show the strong dependence on the effective temperature that we observed in the LTE nitrogen abundances. When the NLTE nitrogen and carbon abundances of the A3-F0 supergiants are compared to those of the main-sequence B stars, we find [log epsilon(N/C)(AI) - log epsilon(N/C)(B+)] = +0.38 +/- 0.26. This value is significantly less than the first dredge-up abundances (similar to +0.65 for 10 M. stars) predicted by several evolution scenarios. However, the nonzero [N/C] ratio suggests that the A-type supergiants have undergone some partial mixing of CN-cycled gas. This is similar to recent abundance results for some B-type supergiants (Gies & Lambert 1992; Lennon 1994), suggesting that partial mixing may occur near the main-sequence (possibly by turbulent diffusive mixing; Maeder 1987; Denissenkov 1994). We conclude that the 5-20 M. A-type supergiants in the Galaxy have evolved directly from the main sequence.