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We present a Bayesian inference on the neutral hydrogen fraction of the intergalactic medium (IGM), $\overline{x}_{\small HI}$, at z ∼ 6–8 using the properties of Lyman break galaxies (LBGs) during the epoch of reionization. We use large samples of LBG candidates at 5.5 ≤ z ≤ 8.2 with spectroscopy from Keck/DEIMOS and Keck/MOSFIRE. For each galaxy, we incorporate either the Lyman-α (Lyα) equivalent width (EW) for detections or the EW limit spectrum for non-detections to parametrize the EW distribution at various ultraviolet brightnesses for a given redshift. Using our reference sample of galaxy candidates from the ionized universe at z ∼ 6.0, we are able to infer $\overline{x}_{\small HI}$ at two redshifts: z ∼ 6.7 and z ∼ 7.6. This work includes intrinsically faint, gravitationally lensed galaxies at z ∼ 6.0 in order to constrain the intrinsic faint-end Lyα EW distribution and provide a comparable population of galaxies to counterparts in our sample that are at higher redshift. The inclusion of faint galaxy candidates, in addition to a more sophisticated modelling framework, allows us to better isolate effects of the interstellar medium and circumgalactic medium on the observed Lyα distribution from those of the IGM. We infer an upper limit of $\overline{x}_{\small HI}$ ≤ 0.25 (0.44) at z = 6.7 ± 0.2 and a neutral fraction of $\overline{x}_{\small HI}$ = $0.83^{+0.08}_{-0.11}$ (0.83$^{+0.11}_{-0.21}$) at z = 7.6 ± 0.6, both within 68 per cent (95 per cent) uncertainty, results that favour a moderately late and fairly rapid reionization.