  We use a large sample of galaxies at z~3 to establish a relationship between
reddening, neutral gas covering fraction (fcov(HI)), and the escape of ionizing
photons at high redshift. Our sample includes 933 galaxies at z~3, 121 of which
have very deep spectroscopic observations (>7 hrs) in the rest-UV
(lambda=850-1300 A) with Keck/LRIS. Based on the high covering fraction of
outflowing optically-thick HI indicated by the composite spectra of these
galaxies, we conclude that photoelectric absorption, rather than dust
attenuation, dominates the depletion of ionizing photons. By modeling the
composite spectra as the combination of an unattenuated stellar spectrum
including nebular continuum emission with one that is absorbed by HI and
reddened by a line-of-sight extinction, we derive an empirical relationship
between E(B-V) and fcov(HI). Galaxies with redder UV continua have larger
covering fractions of HI characterized by higher line-of-sight extinctions. Our
results are consistent with the escape of Lya through gas-free lines-of-sight.
Covering fractions based on low-ionization interstellar absorption lines
systematically underpredict those deduced from the HI lines, suggesting that
much of the outflowing gas may be metal-poor. We develop a model which connects
the ionizing escape fraction with E(B-V), and which may be used to estimate the
escape fraction for an ensemble of high-redshift galaxies. Alternatively,
direct measurements of the escape fraction for our data allow us to constrain
the intrinsic 900-to-1500 A flux density ratio to be >0.20, a value that favors
stellar population models that include weaker stellar winds, a flatter initial
mass function, and/or binary evolution. Lastly, we demonstrate how the
framework discussed here may be used to assess the pathways by which ionizing
radiation escapes from high-redshift galaxies. [Abridged]
