Cane toads (Bufo marinus) are large anurans (weighing up to 2 kg) that were introduced to Australia 70 years ago to control insect pests in sugar-cane fields. But the result has been disastrous because the toads are toxic and highly invasive. Here we show that the annual rate of progress of the toad invasion front has increased about fivefold since the toads first arrived; we find that toads with longer legs can not only move faster and are the first to arrive in new areas, but also that those at the front have longer legs than toads in older (long-established) populations. The disaster looks set to turn into an ecological nightmare because of the negative effects invasive species can have on native ecosystems; over many generations, rates of invasion will be accelerated owing to rapid adaptive change in the invader, with continual 'spatial selection' at the expanding front favouring traits that increase the toads' dispersal.