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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 ∼1 My before and ended ∼1.5 My after the major impact events that produced the large Popigai and Chesapeake Bay craters ∼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.