The mechanisms underlying ribonucleoprotein (RNP) granule assembly, including the basis for establishing and maintaining RNP granules with distinct composition, are unknown. One prominent type of RNP granule is the stress granule (SG), a dynamic and reversible cytoplasmic assembly formed in eukaryotic cells in response to stress. Here, we show that SGs assemble through liquid-liquid phase separation (LLPS) arising from interactions distributed unevenly across a core protein-RNA interaction network. The central node of this network is G3BP1, which functions as a molecular switch that triggers RNA-dependent LLPS in response to a rise in intracellular free RNA concentrations. Moreover, we show that interplay between three distinct intrinsically disordered regions (IDRs) in G3BP1 regulates its intrinsic propensity for LLPS, and this is fine-tuned by phosphorylation within the IDRs. Further regulation of SG assembly arises through positive or negative cooperativity by extrinsic G3BP1-binding factors that strengthen or weaken, respectively, the core SG network.