Viruses are highly evolvable, but what traits endow this property? The high mutation rates of viruses certainly play a role, but factors that act above the genetic code, like protein thermostability, are also expected to contribute. We studied how the thermostability of a model virus, bacteriophage λ, affects its ability to evolve to use a new receptor, a key evolutionary transition that can cause host-range evolution. Using directed evolution and synthetic biology techniques we generated a library of host-recognition protein variants with altered stabilities and then tested their capacity to evolve to use a new receptor. Variants fell within three stability classes: stable, unstable, and catastrophically unstable. The most evolvable were the two unstable variants, whereas seven of eight stable variants were significantly less evolvable, and the two catastrophically unstable variants could not grow. The slowly evolving stable variants were delayed because they required an additional destabilizing mutation. These results are particularly noteworthy because they contradict a widely supported contention that thermostabilizing mutations enhance evolvability of proteins by increasing mutational robustness. Our work suggests that the relationship between thermostability and evolvability is more complex than previously thought, provides evidence for a new molecular model of host-range expansion evolution, and identifies instability as a potential predictor of viral host-range evolution.