Interrogating physical processes that occur within the lowermost mantle is a key to understanding Earth's evolution and present-day inner composition. Among such processes, partial melting has been proposed to explain mantle regions with ultralow seismic velocities near the core-mantle boundary, but experimental validation at the appropriate temperature and pressure regimes remains challenging. Using laser-heated diamond anvil cells, we constructed the solidus curve of a natural fertile peridotite between 36 and 140 gigapascals. Melting at core-mantle boundary pressures occurs at 4180 +/- 150 kelvin, which is a value that matches estimated mantle geotherms. Molten regions may therefore exist at the base of the present-day mantle. Melting phase relations and element partitioning data also show that these liquids could host many incompatible elements at the base of the mantle.