The 'Late Heavy Bombardment' was a phase in the impact history of the Moon that occurred 3.8 4.0 Gyr ago, when the lunar basins with known dates were formed. But no record of this event has yet been reported from the few surviving rocks of this age on the Earth. Here we report tungsten isotope anomalies, based on the (182)Hf (182)W system (half-life of 9 Myr), in metamorphosed sedimentary rocks from the 3.7 3.8-Gyr-old Isua greenstone belt of West Greenland and closely related rocks from northern Labrador, Canada. As it is difficult to conceive of a mechanism by which tungsten isotope heterogeneities could have been preserved in the Earth's dynamic crust mantle environment from a time when short-lived (182)Hf was still present, we conclude that the metamorphosed sediments contain a component derived from meteorites.