We present a new method to directly measure the opacity from H I Lyman limit (LL) absorption κ_LL along quasar sight lines by the intergalactic medium (IGM). The approach analyzes the average ("stacked") spectrum of an ensemble of quasars at a common redshift to infer the mean free path λ⁹¹² _mfp to ionizing radiation. We apply this technique to 1800 quasars at z = 3.50-4.34 drawn from the Sloan Digital Sky Survey (SDSS), giving the most precise measurements on κ_LL at any redshift. From z = 3.6 to 4.3, the opacity increases steadily as expected and is well parameterized by λ⁹¹² _mfp = λ₀ – b _λ(z – 3.6) with λ₀ = (48.4 ± 2.1) h ^–1 ₇₂ Mpc and b _λ = (38.0 ± 5.3) h ^–1 ₇₂ Mpc (proper distance). The relatively high λ⁹¹² _mfp values indicate that the incidence of systems which dominate κ_LL evolves less strongly at z > 3 than that of the Lyα forest. We infer a mean free path three times higher than some previous estimates, a result which has important implications for the photoionization rate derived from the emissivity of star-forming galaxies and quasars. Finally, our analysis reveals a previously unreported, systematic bias in the SDSS quasar sample related to the survey's color targeting criteria. This bias potentially affects all z ~ 3 IGM studies using the SDSS database.