We have continued our studies of the Classical Nova outburst by evolving TNRs on 1.25Msun and 1.35Msun WDs (ONeMg composition) under conditions which produce mass ejection and a rapid increase in the emitted light, by examining the effects of changes in the nuclear reaction rates on both the observable features and the nucleosynthesis during the outburst. In order to improve our calculations over previous work, we have incorporated a modern nuclear reaction network into our hydrodynamic computer code. We find that the updates in the nuclear reaction rate libraries change the amount of ejected mass, peak luminosity, and the resulting nucleosynthesis. In addition, as a result of our improvements, we discovered that the pep reaction was not included in our previous studies of CN explosions. Although the energy production from this reaction is not important in the Sun, the densities in WD envelopes can exceed $10^4$ gm cm$^{-3}$ and the presence of this reaction increases the energy generation during the time that the p-p chain is operating. The effect of the increased energy generation is to reduce the evolution time to the peak of the TNR and, thereby, the accreted mass as compared to the evolutionary sequences done without this reaction included. As expected from our previous work, the reduction in accreted mass has important consequences on the characteristics of the resulting TNR and is discussed in this paper.