The Red Queen describes a view of nature in which species continually evolve but do not become better adapted. It is one of the more distinctive metaphors of evolutionary biology, but no test of its claim that speciation occurs at a constant rate has ever been made against competing models that can predict virtually identical outcomes, nor has any mechanism been proposed that could cause the constant-rate phenomenon. Here we use 101 phylogenies of animal, plant and fungal taxa to test the constant-rate claim against four competing models. Phylogenetic branch lengths record the amount of time or evolutionary change between successive events of speciation. The models predict the distribution of these lengths by specifying how factors combine to bring about speciation, or by describing how rates of speciation vary throughout a tree. We find that the hypotheses that speciation follows the accumulation of many small events that act either multiplicatively or additively found support in 8% and none of the trees, respectively. A further 8% of trees hinted that the probability of speciation changes according to the amount of divergence from the ancestral species, and 6% suggested speciation rates vary among taxa. By comparison, 78% of the trees fit the simplest model in which new species emerge from single events, each rare but individually sufficient to cause speciation. This model predicts a constant rate of speciation, and provides a new interpretation of the Red Queen: the metaphor of species losing a race against a deteriorating environment is replaced by a view linking speciation to rare stochastic events that cause reproductive isolation. Attempts to understand species-radiations or why some groups have more or fewer species should look to the size of the catalogue of potential causes of speciation shared by a group of closely related organisms rather than to how those causes combine.