  We present the results of rest-frame ultraviolet spectroscopic observations
of a sample of 14 z~3 star-forming galaxies in the SSA22a field. These spectra
are characterized by unprecedented depth in the Lyman-continuum region. For the
first time, we have detected escaping ionizing radiation from individual
galaxies at high redshift, with two of the 14 objects showing significant
emission below the Lyman limit. We also measured the ratio of emergent flux
density at 1500 AA to that in the Lyman-continuum region, for the individual
detections (C49 and D3) and the sample average. If a correction for the average
IGM opacity is applied to the spectra of the objects C49 and D3, we find
f_1500/f_900,corr(C49)=4.5 and f_1500/f_900,corr(D3)=2.9. The average emergent
flux-density ratio in our sample is <f_1500/f_900,corr>=22, implying an escape
fraction ~4.5 times lower than inferred from the composite spectrum in Steidel
et al. (2001). If this new estimate is representative of LBGs, their
contribution to the metagalactic ionizing radiation field is
J_nu(900)~2.6x10^{-22} erg/s/cm^2/Hz/sr, comparable to the contribution of
optically-selected quasars at the same redshift. The sum of the contributions
from galaxies and quasars is consistent with recent estimates of the level of
the ionizing background at z~3, inferred from the HI Ly-alpha forest optical
depth. There is significant variance among the emergent far-UV spectra in our
sample, yet the factors controlling the detection or non-detection of
Lyman-continuum emission from galaxies are not well-determined. Because we do
not yet understand the source of this variance, significantly larger samples
will be required to obtain robust constraints on the galaxy contribution to the
ionizing background at z~3 and beyond (abridged).
