The CRISPR system in bacteria and archaea provides adaptive immunity against mobile genetic elements. Type III CRISPR systems detect viral RNA, resulting in the activation of two regions of the Cas10 protein: an HD nuclease domain (which degrades viral DNA)(1,2) and a cyclase domain (which synthesizes cyclic oligoadenylates from ATP)(3-5). Cyclic oligoadenylates in turn activate defence enzymes with a CRISPR-associated Rossmann fold domain(6), sculpting a powerful antiviral response(7-10) that can drive viruses to extinction(7,8). Cyclic nucleotides are increasingly implicated in host-pathogen interactions(11-13). Here we identify a new family of viral anti-CRISPR (Acr) enzymes that rapidly degrade cyclic tetra-adenylate (cA(4)). The viral ring nuclease AcrIII-1 is widely distributed in archaeal and bacterial viruses and in proviruses. The enzyme uses a previously unknown fold to bind cA(4) specifically, and a conserved active site to rapidly cleave this signalling molecule, allowing viruses to neutralize the type III CRISPR defence system. The AcrIII-1 family has a broad host range, as it targets cA(4) signalling molecules rather than specific CRISPR effector proteins. Our findings highlight the crucial role of cyclic nucleotide signalling in the conflict between viruses and their hosts. Bacteria and archaea use cyclic oligoadenylate molecules as part of the CRISPR system for antiviral defence; here, a family of viral enzymes that rapidly degrades cyclic oligoadenylates is identified and biochemically and structurally described.