We present results of a deep spectroscopic survey designed to quantify the statistics of the escape of ionizing photons from star-forming galaxies at z~3. We measure the ratio of ionizing to non-ionizing UV flux density <f900/f1500>_obs, where f900 is the mean flux density evaluated over the range 880,910 A. We quantify the emergent ratio of ionizing to non-ionizing UV flux density by analyzing high-S/N composite spectra formed from sub-samples with common observed properties and numbers sufficient to reduce the statistical uncertainty in the modeled IGM+CGM correction to obtain precise values of <f900/f1500>_out, including a full-sample average <f900/f1500>_out=\(0.057\pm0.006\). We further show that <f900/f1500>_out increases monotonically with Ly\(\alpha\) rest equivalent width, inducing an inverse correlation with UV luminosity as a by-product. We fit the composite spectra using stellar spectral synthesis together with models of the ISM in which a fraction f_c of the stellar continuum is covered by gas with column density N(HI). We show that the composite spectra simultaneously constrain the intrinsic properties of the stars (L900/L1500)_int along with f_c, N(HI), E(B-V), and \(f_{esc,abs}\), the absolute escape fraction of ionizing photons. We find a sample-averaged \(f_{esc,abs} =0.09\pm0.01\), and that subsamples fall along a linear relation \(\langle f_{esc,abs}\rangle \sim 0.75W(Ly\alpha)/110 A\). We use the FUV luminosity function, the distribution function \(nW(Ly\alpha)\), and the relationship between \(W(Ly\alpha)\) and <f900/f1500>_out to estimate the total ionizing emissivity of \(z\sim3\) star-forming galaxies with Muv < -19.5: \(\epsilon_{LyC}\sim 6\times10^{24}\) ergs/s/Hz/Mpc\(^3\), exceeding the contribution of QSOs by a factor of \(\sim 3\), and accounting for \(\sim50\)% of the total \(\epsilon_{LyC}\) at \(z\sim3\) estimated using indirect methods.