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.