The distribution of species body size within taxonomic groups exhibits a heavy right tail extending over many orders of magnitude, where most species are much larger than the smallest species. We provide a simple model of cladogenetic diffusion over evolutionary time that omits explicit mechanisms for interspecific competition and other microevolutionary processes, yet fully explains the shape of this distribution. We estimate the model's parameters from fossil data and find that it robustly reproduces the distribution of 4002 mammal species from the late Quaternary. The observed fit suggests that the asymmetric distribution arises from a fundamental trade-off between the short-term selective advantages (Cope's rule) and long-term selective risks of increased species body size in the presence of a taxon-specific lower limit on body size.