Biomolecular condensates organize biochemistry in time and space, yet little is known about how cells control either the position or scale of these assemblies. In cells, condensates often appear as dispersed, relatively small assemblies that do not grow (coarsen) into a single droplet despite their propensity to coalesce. Here we report that ribonucleoprotein condensates of the Q-rich protein Whi3 interact with the endoplasmic reticulum, prompting us to hypothesize that membrane association controls the position and size of condensates. Reconstitution of Whi3 condensates on supported lipid bilayers reveals that association with a diffusive lipid surface promotes condensation at both physiological ionic strength and protein concentration. Notably, these assemblies rapidly arrest, matching size distributions seen in cells. The timing of the arrest is influenced by the ordering of protein-protein and protein-RNA interactions and controlled by the slow diffusion of complexes induced by the membrane. This slowed diffusion limits both transfer of small protein-RNA complexes between condensates and their coalescence, thus driving coarsening to arrest. Our experiments reveal a tradeoff between locally-enhanced protein concentration at membranes, which favors condensation, and an accompanying reduction in diffusion, which restricts coarsening. Thus, membranes can maintain a population of small condensates in the absence of active mechanisms. Given that many condensates are bound to endomembranes, we predict that the biophysical properties of lipid bilayers are key for controlling condensate sizes throughout the cell.One sentence summaryAssembly on a membrane surface positions and scales biomolecular condensates by controlling relative diffusion rates of proteins and nucleic acids.