Ecological models show that complexity usually destabilizes food webs [1,2], predicting that food webs should not amass the large numbers of interacting species that are in fact found in nature *RF 3-5*. Here, using nonlinear models, we study the influence of interaction strength (likelihood of consumption of one species by another) on food-web dynamics away from equilibrium. Consistent with previous suggestions [1,6], our results show that weak to intermediate strength links are important in promoting community persistence and stability. Weak links act to dampen oscillations between consumers and resources. This tends to maintain population densities further away from zero, decreasing the statistical chance that a population will become extinct (lower population densities are more prone to such chances). Data on interaction strengths in natural food webs [7-11] indicate that food-web interaction strengths are indeed characterized by many weak interactions and a few strong interactions.