The fraction of ionized hydrogen left over from the Big Bang provides evidence for the time of formation of the first stars and quasar black holes in the early Universe; such objects provide the high-energy photons necessary to ionize hydrogen. Spectra of the two most distant known quasars show nearly complete absorption of photons with wavelengths shorter than the Lyman alpha transition of neutral hydrogen, indicating that hydrogen in the intergalactic medium (IGM) had not been completely ionized at a redshift of z approximately 6.3, about one billion years after the Big Bang. Here we show that the IGM surrounding these quasars had a neutral hydrogen fraction of tens of per cent before the quasar activity started, much higher than the previous lower limits of approximately 0.1 per cent. Our results, when combined with the recent inference of a large cumulative optical depth to electron scattering after cosmological recombination therefore suggest the presence of a second peak in the mean ionization history of the Universe.