Broad absorption signatures from alkali metals, such as the sodium (Na I) and potassium (K I) resonance doublets, have long been predicted in the optical atmospheric spectra of cloud-free irradiated gas giant exoplanets(1-3). However, observations have revealed only the narrow cores of these features rather than the full pressure-broadened profiles(4-6). Cloud and haze opacity at the day-night planetary terminator are considered to be responsible for obscuring the absorption-line wings, which hinders constraints on absolute atmospheric abundances(7-9). Here we report an optical transmission spectrum for the 'hot Saturn' exoplanet WASP-96b obtained with the Very Large Telescope, which exhibits the complete pressure-broadened profile of the sodium absorption feature. The spectrum is in excellent agreement with cloud-free, solar-abundance models assuming chemical equilibrium. We are able to measure a precise, absolute sodium abundance of log epsilon(Na) = 6.9(-0.4)(+0.6), and use it as a proxy for the planet's atmospheric metallicity relative to the solar value (Z(p)/Z(circle dot) = 2.3(-1.7)(+8.9)). This result is consistent with the mass-metallicity trend observed for Solar System planets and exoplanets(10-12).