Cosmic rays are the highest-energy particles found in nature. Measurements of the mass composition of cosmic rays with energies of 1017–1018 electronvolts are essential to understanding whether they have galactic or extragalactic sources. It has also been proposed that the astrophysical neutrino signal1 comes from accelerators capable of producing cosmic rays of these energies2 . Cosmic rays initiate air showers—cascades of secondary particles in the atmosphere—and their masses can be inferred from measurements of the atmospheric depth of the shower maximum3 (Xmax; the depth of the air shower when it contains the most particles) or of the composition of shower particles reaching the ground4 . Current measurements5 have either high uncertainty, or a low duty cycle and a high energy threshold. Radio detection of cosmic rays6–8 is a rapidly developing technique9 for determining Xmax (refs 10, 11) with a duty cycle of, in principle, nearly 100 per cent. The radiation is generated by the separation of relativistic electrons and positrons in the geomagnetic field and a negative charge excess in the shower front6,12. Here we report radio measurements of Xmax with a mean uncertainty of 16 grams per square centimetre for air showers initiated by cosmic rays with energies of 1017–1017.5 electronvolts. This high resolution in Xmax enables us to determine the mass spectrum of the cosmic rays: we find a mixed composition, with a light-mass fraction (protons and helium nuclei) of about 80 per cent. Unless, contrary to current expectations, the extragalactic component of cosmic rays contributes substantially to the total flux below 1017.5 electronvolts, our measurements indicate the existence of an additional galactic component, to account for the light composition that we measured in the 1017–1017.5 electronvolt range. Observations were made with the Low Frequency Array (LOFAR13), a radio telescope consisting of thousands of crossed dipoles with built-in air-shower-detection capability14. LOFAR continuously records the radio signals from air showers, while simultaneously running astronomical observations. It comprises a scintillator array (LORA) that triggers the read-out of buffers, storing the full waveforms received by all antennas