  Arguably the best hope of understanding the tail end of the reionization of
the intergalactic medium (IGM) at redshift z > 6 is through the detection and
characterization of the Gunn-Peterson (GP) damping wing absorption of the IGM
in bright quasar spectra. However, the use of quasar spectra to measure the IGM
damping wing requires a model of the quasar's intrinsic Lyman-alpha emission
line. Here we quantify the uncertainties in the intrinsic line shapes, and how
those uncertainties affect the determination of the IGM neutral fraction. We
have assembled a catalog of high-resolution HST spectra of the emission lines
of unobscured low-redshift quasars, and have characterized the variance in the
shapes of their lines. We then add simulated absorption from the high-redshift
IGM to these quasar spectra in order to determine the corresponding
uncertainties in reionization constraints using current and future samples of z
> 6 quasar spectra. We find that, if the redshift of the Lyman-alpha emission
line is presumed to coincide with the systemic redshift determined from metal
lines, the inferred IGM neutral fraction is systematically biased to low values
due to a systematic blueshift of the Lyman-alpha line relative to the metal
lines. If a similar blueshift persists in quasars at z > 6, this bias
strengthens previous claims of a significant neutral hydrogen fraction at z ~
6. This technique is capable of making a robust distinction between a highly
ionized (x_IGM ~ 10^-3) and a neutral (x_IGM = 1) IGM with even a few bright
quasars.
