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(1). 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(2). One explanation for this overabundance is that a 'late veneer' of primitive material was added to the bulk silicate Earth after the core formed(2). To test this hypothesis, tungsten isotopes are useful for two reasons: first, because the late veneer material had a different W-182/W-184 ratio to that of the bulk silicate Earth, and second, proportionally more material was added to the Earth than to the Moon(3). Thus, if a late veneer did occur, the bulk silicate Earth and the Moon must have different W-182/W-184 ratios. Moreover, the Moon-forming impact would also have created W-182 differences because the mantle and core material of the impactor with distinct W-182/W-184 would have mixed with the proto-Earth during the giant impact. However the W-182/W-184 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 W-182 excess of 27 +/- 4 parts per million over the present-day bulk silicate Earth. This excess is consistent with the expected W-182 difference resulting from a late veneer with a total mass and composition inferred from HSE systematics(2). 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 W-182 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 W-182 of the pre-late-veneer bulk silicate Earth and the Moon is enigmatic and constitutes a challenge to current models of lunar origin.