Measles virus (MV) deficient in C protein (Cko) expression efficiently induces both stress granules (SG) and interferon (IFNβ), whereas isogenic wild-type (WT) and V mutant (Vko) viruses do not. We therefore examined the effect of IFNβ pretreatment on SG formation, and the roles played by the IFN-inducible double-stranded (ds) RNA-dependent protein kinase (PKR) and dsRNA adenosine deaminase (ADAR1). SG formation in ADAR1-sufficient cells infected with WT or Vko mutant virus was enhanced by IFN treatment and was PKR-dependent. SG formation in Cko virus-infected cells was already high without IFN treatment and was not further enhanced by IFN. IFN treatment alone, in the absence of infection, induced SG formation in ADAR1-deficient but not ADAR1-sufficient cells. Type I IFN-induced enhancement in SG formation occurred by a canonical IFN signaling response dependent upon STAT1 and STAT2. These results further establish ADAR1 as a suppressor of the interferon and SG innate immune responses.