The terrestrial and lunar cratering rate is often assumed to have been nearly constant over the past 3Gyr. Different lines of evidence, however, suggest that the impact flux from kilometre-sized bodies increased by at least a factor of two over the long-term average during the past ~100Myr. Here we argue that this apparent surge was triggered by the catastrophic disruption of the parent body of the asteroid Baptistina, which we infer was a ~170-km-diameter body (carbonaceous-chondrite-like) that broke up Myr ago in the inner main asteroid belt. Fragments produced by the collision were slowly delivered by dynamical processes to orbits where they could strike the terrestrial planets. We find that this asteroid shower is the most likely source (>90 per cent probability) of the Chicxulub impactor that produced the Cretaceous/Tertiary (K/T) mass extinction event 65Myr ago.