Water is perhaps the most important molecule in the solar system, and determining its origin and distribution in planetary interiors has important implications for understanding the evolution of planetary bodies. Here we report in situ measurements of the isotopic composition of hydrogen dissolved in primitive volcanic glass and olivine-hosted melt inclusions recovered from the Moon by the Apollo 15 and 17 missions. After consideration of cosmic-ray spallation and degassing processes, our results demonstrate that lunar magmatic water has an isotopic composition that is indistinguishable from that of the bulk water in carbonaceous chondrites and similar to that of terrestrial water, implying a common origin for the water contained in the interiors of Earth and the Moon.