  A bright quasar residing in a dense and largely neutral intergalactic medium
(IGM) at high redshifts (z > 6) will be surrounded by a large cosmological
Stromgren sphere. The quasar's spectrum will then show a sharp increase in
resonant Lyman line absorption at wavelengths approaching and shorter than that
corresponding to the Stromgren sphere's boundary along the line of sight. We
show here that simultaneously considering the measured absorption in two or
more hydrogen Lyman lines can provide the dynamical range required to detect
this feature. We model broad and robust features of the Lyman alpha and Lyman
beta regions of the spectrum of the z=6.28 quasar SDSS J1030+0524, using a
hydrodynamical simulation. From the steep wavelength-dependence of the inferred
absorption opacity, we detect the boundary of the Stromgren sphere at a proper
distance of 6.0 +/- 0.2 Mpc away from the source redshift. From the spectrum
alone, we also find that beyond this distance, cosmic hydrogen turns nearly
neutral, with a neutral fraction of x_HI > 0.2, and that the ionizing
luminosity of this quasar is in the range (5.2 +/- 2.5) times 10^{56}
photons/sec. The method presented here, when applied to future quasars, can
probe the complex topology of overlapping ionized regions, and can be used to
study the details of the reionization process.
