We present the results of rest-frame UV spectroscopic observations of a sample of 14 z ~ 3 star-forming galaxies in the SSA 22a 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 2 of the 14 objects showing significant emission below the Lyman limit. We also measured the ratio of emergent flux density at 1500 Å 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 = 22, implying an escape fraction ~4.5 times lower than inferred from the composite spectrum from Steidel and coworkers. If this new estimate is representative of LBGs, their contribution to the metagalactic ionizing radiation field is J_ν(900) ~ 2.6 × 10^(-22) ergs s^(-1) cm^(-2) Hz^(-1) sr^(-1), 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 H I Lyα forest optical depth. There is significant variance among the emergent far-UV spectra in our sample, yet the factors controlling the detection or nondetection 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
