Approximately one-third of global CO2 fixation occurs in a phase separated algal organelle called the pyrenoid. Existing data suggest that the pyrenoid forms by the phase-separation of the CO2-fixing enzyme Rubisco with a linker protein; however, the molecular interactions underlying this phase-separation remain unknown. Here we present the structural basis of the interactions between Rubisco and its intrinsically disordered linker protein EPYC1 (Essential Pyrenoid Component 1) in the model alga Chlamydomonas reinhardtii. We find that EPYC1 consists of five evenly-spaced Rubisco-binding regions that share sequence similarity. Single-particle cryo-electron microscopy of one of these regions in complex with Rubisco indicates that each Rubisco holoenzyme has eight binding sites for EPYC1, one on each Rubisco small subunit. Interface mutations disrupt binding, phase separation, and pyrenoid formation. Cryo-electron tomography supports a model where EPYC1 and Rubisco form a co-dependent multivalent network of specific low-affinity bonds, giving the matrix liquid-like properties. Our results advance the structural and functional understanding of the phase separation underlying the pyrenoid, an organelle that plays a fundamental role in the global carbon cycle.