Cosmic rays are the highest-energy particles found in nature. Measurements of the mass composition of cosmic rays with energies of 10(17)-10(18) 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 energies(2). 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 ground(4). Current measurements(5) have either high uncertainty, or a low duty cycle and a high energy threshold. Radio detection of cosmic rays(6-8) is a rapidly developing technique(9) 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 front(6,12). Here we report radio measurements of X-max 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 10(17.5) electronvolts, our measurements indicate the existence of an additional galactic component, to account for the light composition that we measured in the 10(17)-10(17.5) electronvolt range.