Despite advances in theory and experiments, how biodiversity influences the structure and functioning of natural ecosystems remains debated. By applying new theory to data on 84,695 plant, animal, and protist assemblages, we show that the general positive effect of species richness on stocks of biomass, as well as much of the variation in the strength and sign of this effect, is predicted by a fundamental macroecological quantity: the scaling of species abundance with body mass. Standing biomass increases with richness when large-bodied species are numerically rare but is independent of richness when species size and abundance are uncoupled. These results suggest a new fundamental law in the structure of ecological communities and show that the impacts of changes in species richness on biomass are predictable.