The intergalactic medium was not completely reionized until approximately a billion years after the Big Bang, as revealed by observations of quasars with redshifts of less than 6.5. It has been difficult to probe to higher redshifts, however, because quasars have historically been identified in optical surveys, which are insensitive to sources at redshifts exceeding 6.5. Here we report observations of a quasar (ULAS J112001.48+064124.3) at a redshift of 7.085, which is 0.77 billion years after the Big Bang. ULAS J1120+0641 has a luminosity of 6.3 × 1013L⊙ and hosts a black hole with a mass of 2 × 109M⊙ (where L⊙ and M⊙ are the luminosity and mass of the Sun). The measured radius of the ionized near zone around ULAS J1120+0641 is 1.9 megaparsecs, a factor of three smaller than is typical for quasars at redshifts between 6.0 and 6.4. The near-zone transmission profile is consistent with a Lyα damping wing, suggesting that the neutral fraction of the intergalactic medium in front of ULAS J1120+0641 exceeded 0.1.