Many precious, 'iron-loving' metals, such as gold, are surprisingly abundant in the accessible parts of the Earth, given the efficiency with which core formation should have removed them to the planet's deep interior. One explanation of their over-abundance is a 'late veneer'--a flux of meteorites added to the Earth after core formation as a 'terminal' bombardment that culminated in the cratering of the Moon. Some 3.8 billion-year-old rocks from Isua, Greenland, are derived from sources that retain an isotopic memory of events pre-dating this cataclysmic meteorite shower. These Isua samples thus provide a window on the composition of the Earth before such a late veneer and allow a direct test of its importance in modifying the composition of the planet. Using high-precision (less than 6 parts per million, 2 standard deviations) tungsten isotope analyses of these rocks, here we show that they have a isotopic tungsten ratio (182)W/(184)W that is significantly higher (about 13 parts per million) than modern terrestrial samples. This finding is in good agreement with the expected influence of a late veneer. We also show that alternative interpretations, such as partial remixing of a deep-mantle reservoir formed in the Hadean eon (more than four billion years ago) or core-mantle interaction, do not explain the W isotope data well. The decrease in mantle (182)W/(184)W occurs during the Archean eon (about four to three billion years ago), potentially on the same timescale as a notable decrease in (142)Nd/(144)Nd (refs 3 and 6). We speculate that both observations can be explained if late meteorite bombardment triggered the onset of the current style of mantle convection.