Natural selection can promote or hinder a population’s evolvability—the ability to evolve new and adaptive phenotypes—but the underlying mechanisms are poorly understood. To examine how the strength of selection affects evolvability, we subjected populations of yellow fluorescent protein to directed evolution under different selection regimes and then evolved them toward the new phenotype of green fluorescence. Populations under strong selection for the yellow phenotype evolved the green phenotype most rapidly. They did so by accumulating mutations that increase both robustness to mutations and foldability. Under weak selection, neofunctionalizing mutations rose to higher frequency at first, but more frequent deleterious mutations undermined their eventual success. Our experiments show how selection can enhance evolvability by enhancing robustness and create the conditions necessary for evolutionary success.