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 (X_max; 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 rays^{6, 7, 8} is a rapidly developing technique⁹ for determining X_max (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 10¹⁷–10^17.5 electronvolts. This high resolution in X_max 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 10¹⁷–10^17.5 electronvolt range.