According to the most widely accepted theory of lunar origin, a giant impact on the Earth led to the formation of the Moon, and also initiated the final stage of the formation of the Earth's core. Core formation should have removed the highly siderophile elements (HSE) from Earth's primitive mantle (that is, the bulk silicate Earth), yet HSE abundances are higher than expected. One explanation for this overabundance is that a 'late veneer' of primitive material was added to the bulk silicate Earth after the core formed. To test this hypothesis, tungsten isotopes are useful for two reasons: first, because the late veneer material had a different (182)W/(184)W ratio to that of the bulk silicate Earth, and second, proportionally more material was added to the Earth than to the Moon. Thus, if a late veneer did occur, the bulk silicate Earth and the Moon must have different (182)W/(184)W ratios. Moreover, the Moon-forming impact would also have created (182)W differences because the mantle and core material of the impactor with distinct (182)W/(184)W would have mixed with the proto-Earth during the giant impact. However the (182)W/(184)W of the Moon has not been determined precisely enough to identify signatures of a late veneer or the giant impact. Here, using more-precise measurement techniques, we show that the Moon exhibits a (182)W excess of 27 ± 4 parts per million over the present-day bulk silicate Earth. This excess is consistent with the expected (182)W difference resulting from a late veneer with a total mass and composition inferred from HSE systematics. Thus, our data independently show that HSE abundances in the bulk silicate Earth were established after the giant impact and core formation, as predicted by the late veneer hypothesis. But, unexpectedly, we find that before the late veneer, no (182)W anomaly existed between the bulk silicate Earth and the Moon, even though one should have arisen through the giant impact. The origin of the homogeneous (182)W of the pre-late-veneer bulk silicate Earth and the Moon is enigmatic and constitutes a challenge to current models of lunar origin.