We present the discovery and subsequent spectroscopy with Gemini-North of the optical afterglow of the Swift gamma-ray burst ( GRB ) 140515A . The spectrum exhibits a well-detected continuum at wavelengths longer than 8915 Å with a steep decrement to zero flux blueward of 8910 Å due to Ly \alpha absorption at redshift z \approx 6.33 . Some transmission through the Ly \alpha forest is present at 5.2 < z < 5.733 , but none is detected at higher redshift , consistent with previous measurements from quasars and GRB 130606A . We model the red damping wing of Ly \alpha in three ways that provide equally good fits to the data : ( a ) a single host galaxy absorber at z = 6.327 with \log ( N _ { \mathrm { HI } } , \mathrm { cm } ^ { -2 } ) =18.62 \pm 0.08 ; ( b ) pure intergalactic medium ( IGM ) absorption from z = 6.0 to z = 6.328 with a constant neutral hydrogen fraction of \bar { x } _ { \mathrm { HI } } = 0.056 ^ { +0.011 } _ { -0.027 } ; and ( c ) a hybrid model with a host absorber located within an ionized bubble of radius 10 comoving Mpc in an IGM with \bar { x } _ { \mathrm { HI } } = 0.12 \pm 0.05 ( \bar { x } _ { \mathrm { HI } } \lesssim 0.21 at the 2 \sigma level ) . Regardless of the model , the sharpness of the dropoff in transmission is inconsistent with a substantial neutral fraction in the IGM at this redshift . No narrow absorption lines from the host galaxy are detected , indicating a host metallicity of [ Z/H ] \lesssim - 0.8 . Even if we assume that all of the hydrogen absorption is due to the host galaxy , the column is unusually low for a GRB sightline , similar to two out of the other three highest-redshift bursts with measured \log ( N _ { \mathrm { HI } } ) . This is possible evidence that the escape fraction of ionizing photons from normal star-forming galaxies increases at z \gtrsim 6 .