The Moon is thought to have formed from debris ejected by a giant impact with the early 'proto'-Earth(1) and, as a result of the high energies involved, the Moon would have melted to form a magma ocean. The timescales for formation and solidification of the Moon can be quantified by using Hf-182-W-182 and Sm-146-Nd-142 chronometry(2-4), but these methods have yielded contradicting results. In earlier studies(3,5-7), W-182 anomalies in lunar rocks were attributed to decay of Hf-182 within the lunar mantle and were used to infer that the Moon solidified within the first, 60 million years of the Solar System. However, the dominant W-182 component in most lunar rocks reflects cosmogenic production mainly by neutron capture of Ta-181 during cosmic- ray exposure of the lunar surface(3,7), compromising a reliable interpretation in terms of Hf-182-W-182 chronometry. Here we present tungsten isotope data for lunar metals that do not contain any measurable Ta- derived W-182. All metals have identical W-182/W-184 ratios, indicating that the lunar magma ocean did not crystallize within the first, 60 Myr of the Solar System, which is no longer inconsistent with Sm - Nd chronometry(8-11). Our new data reveal that the lunar and terrestrial mantles have identical W-182/W-184. This, in conjunction with Sm-147-Nd-143 ages for the oldest lunar rocks(8-11), constrains the age of the Moon and Earth to 62(-10)(+90) Myr after formation of the Solar System. The identical W-182/W-184 ratios of the lunar and terrestrial mantles require either that the Moon is derived mainly from terrestrial material or that tungsten isotopes in the Moon and Earth's mantle equilibrated in the aftermath of the giant impact, as has been proposed to account for identical oxygen isotope compositions of the Earth and Moon(12).