The crRNA-guided nuclease Cas13 recognizes complementary viral transcripts to trigger the degradation of both host and viral RNA during the type VI CRISPR-Cas antiviral response. How viruses can counteract this immunity is not known. We describe a listeriophage (LS46) encoding an anti-CRISPR protein (AcrVIA1) that inactivated the type VI-A CRISPR system of Listeria seeligeri. Using genetics, biochemistry and structural biology we found that AcrVIA1 interacted with the guide-exposed face of Cas13a, preventing access to the target RNA and the conformational changes required for nuclease activation. Unlike inhibitors of DNA-cleaving Cas nucleases, which cause limited immunosuppression and require multiple infections to bypass CRISPR defenses, a single dose of AcrVIA1 delivered by an individual virion could completely dismantle type VI-A CRISPR-mediated immunity.