Hydrogen is one of the possible alloying elements in the Earth's core, but its siderophile (iron-loving) nature is debated. Here we experimentally examined the partitioning of hydrogen between molten iron and silicate melt at 30-60 gigapascals and 3100-4600 kelvin. We find that hydrogen has a metal/silicate partition coefficient DH ≥ 29 and is therefore strongly siderophile at conditions of core formation. Unless water was delivered only in the final stage of accretion, core formation scenarios suggest that 0.3-0.6 wt% H was incorporated into the core, leaving a relatively small residual H2O concentration in silicates. This amount of H explains 30-60% of the density deficit and sound velocity excess of the outer core relative to pure iron. Our results also suggest that hydrogen may be an important constituent in the metallic cores of any terrestrial planet or moon having a mass in excess of ~10% of the Earth.