Most viruses can infect multiple hosts, yet the selective mechanisms that maintain multi-host generalists over single-host specialists remain an open question. Here we propagate populations of the newly identified bacteriophage øJB01 in coculture with many host genotypes and find that while phage can adapt to infect any of the new hosts, increasing the number of hosts slows the rate of adaptation. We quantify trade-offs in the capacity for individual phage to infect different hosts and find that phage from evolved populations with more hosts are more likely to be generalists. Sequencing of evolved phage reveals strong selection and the genetic basis of adaptation, supporting a model that shows how the addition of more potential hosts to a community can select for low-fitness generalists over high-fitness specialists. Our results show how evolution with multiple hosts alters the rate of viral adaptation and provides empirical support for an evolutionary mechanism that promotes generalists over specialists.