The ubiquity of mechanisms that generate genetic variation has spurred arguments that evolvability, the ability to generate adaptive variation, has itself evolved in response to natural selection. The high mutation rate of RNA viruses is postulated to be an adaptation for evolvability, but the paradox is that whereas some RNA viruses evolve at high rates, others are highly stable. Here we show that evolvability in the RNA bacteriophage phi6 is also determined by the accessibility of advantageous genotypes within the mutational neighbourhood (the set of mutants one or a few mutational steps away). We found that two phi6 populations that were derived from a single ancestral phage repeatedly evolved at different rates and toward different fitness maxima. Fitness measurements of individual phages showed that the fitness distribution of mutants differed between the two populations. Whereas population A, which evolved toward a higher maximum, had a distribution that contained many advantageous mutants, population B, which evolved toward a lower maximum, had a distribution that contained only deleterious mutants. We interpret these distributions to measure the fitness effects of genotypes that are mutationally available to the two populations. Thus, the evolvability of phi6 is constrained by the distribution of its mutational neighbours, despite the fact that this phage has the characteristic high mutation rate of RNA viruses.