Cyclic GMP-AMP synthase (cGAS) is a major sensor of cytosolic DNA from invading pathogens and damaged cellular organelles. Activation of cGAS promotes liquid-like phase separation and formation of membraneless cytoplasmic structures. Here, we found that cGAS bound G3BP1, a double-stranded nucleic acid helicase involved in the formation of stress granules. Loss of G3BP1 blocked subcellular cGAS condensation and suppressed the interferon response to intracellular DNA and DNA virus particles in cells. Furthermore, an RNA-dependent association with PKR promoted G3BP1 foci formation and cGAS-dependent interferon responses. Together, these results indicate that PKR promotes the formation of G3BP1-dependent, membraneless cytoplasmic structures necessary for the DNA-sensing function of cGAS in human cells. These data suggest that there is a previously unappreciated link between nucleic acid sensing pathways, which requires the formation of specialized subcellular structures.