  Cosmic rays are the highest energy particles found in nature. Measurements of
the mass composition of cosmic rays between 10^{17} eV and 10^{18} eV are
essential to understand whether this energy range is dominated by Galactic or
extragalactic sources. It has also been proposed that the astrophysical
neutrino signal comes from accelerators capable of producing cosmic rays of
these energies. Cosmic rays initiate cascades of secondary particles (air
showers) in the atmosphere and their masses are inferred from measurements of
the atmospheric depth of the shower maximum, Xmax, or the composition of shower
particles reaching the ground. Current measurements suffer from either low
precision, or a low duty cycle and a high energy threshold. Radio detection of
cosmic rays is a rapidly developing technique, suitable for determination of
Xmax with a duty cycle of in principle nearly 100%. The radiation is generated
by the separation of relativistic charged particles in the geomagnetic field
and a negative charge excess in the shower front. Here we report radio
measurements of Xmax with a mean precision of 16 g/cm^2 between
10^{17}-10^{17.5} eV. Because of the high resolution in $Xmax we can determine
the mass spectrum and find a mixed composition, containing a light mass
fraction of ~80%. Unless the extragalactic component becomes significant
already below 10^{17.5} eV, our measurements indicate an additional Galactic
component dominating at this energy range.
