Analyses of pelagic limestones indicate that the flux of extraterrestrial helium-3 to Earth was increased for a 2.5-million year (My) period in the late Eocene. The enhancement began approximately 1 My before and ended approximately 1.5 My after the major impact events that produced the large Popigai and Chesapeake Bay craters approximately 36 million years ago. The correlation between increased concentrations of helium-3, a tracer of fine-grained interplanetary dust, and large impacts indicates that the abundance of Earth-crossing objects and dustiness in the inner solar system were simultaneously but only briefly enhanced. These observations provide evidence for a comet shower triggered by an impulsive perturbation of the Oort cloud.