Earlier models of virus evolution during single infections do not include target cell limitation. Here we extend the basic mathematical framework of these theories and study the effect of target cell limitation on the evolution of antigenic variation, increasing replication rates and increasing virus load. We find that target cell limitation provides a selection pressure against antigenic diversification, and can therefore provide a limit to diversity. Antigenic diversity increases virus load; at the maximum level of antigenic diversity, virus load, too, is at a maximum.