Batesian mimicry, in which harmless species (mimics) deter predators by deceitfully imitating the warning signals of noxious species (models), generates striking cases of phenotypic convergence that are classic examples of evolution by natural selection. However, mimicry of venomous coral snakes has remained controversial because of unresolved conflict between the predictions of mimicry theory and empirical patterns in the distribution and abundance of snakes. Here we integrate distributional, phenotypic and phylogenetic data across all New World snake species to demonstrate that shifts to mimetic coloration in nonvenomous snakes are highly correlated with coral snakes in both space and time, providing overwhelming support for Batesian mimicry. We also find that bidirectional transitions between mimetic and cryptic coloration are unexpectedly frequent over both long- and short-time scales, challenging traditional views of mimicry as a stable evolutionary 'end point' and suggesting that insect and snake mimicry may have different evolutionary dynamics.