Food webs have markedly non-random network structure. Ecologists maintain that this non-random structure is key for stability, since large random ecological networks would invariably be unstable and thus should not be observed empirically. Here we show that a simple yet overlooked feature of natural food webs, the correlation between the effects of consumers on resources and those of resources on consumers, substantially accounts for their stability. Remarkably, random food webs built by preserving just the distribution and correlation of interaction strengths have stability properties similar to those of the corresponding empirical systems. Surprisingly, we find that the effect of topological network structure on stability, which has been the focus of countless studies, is small compared to that of correlation. Hence, any study of the effects of network structure on stability must first take into account the distribution and correlation of interaction strengths.