  We report the detection of significant Lyman continuum flux in the composite
spectrum of 29 Lyman break galaxies (LBGs) with redshifts <z> = 3.40+/-0.09.
After correction for opacity due to intervening absorption using a new
composite QSO spectrum evaluated at the same redshift, the ratio of emergent
flux density at 1500 \AA in the rest frame to that in the Lyman continuum is
L(1500)/L(900) = 4.6 +/- 1.0. If the relative intensity of the inferred
escaping Lyman continuum radiation is typical of LBGs at z ~ 3 (the galaxies in
this sample are drawn from the bluest quartile of LBG spectral energy
distributions due to known selection effects), then observed LBGs produce about
5 times more H-ionizing photons per unit co-moving volume than QSOs at z ~ 3.
The associated contribution to the metagalactic ionizing radiation field is
J_{\nu}(912) = 1.2+/-0.3 x 10^{-21} ergs s^{-1} cm^{-2} Hz^{-1} sr^{-1} at z ~
3, very close to most estimates of the radiation field background based on the
``proximity effect''. A preliminary analysis of the density of faint QSOs in
our Lyman break galaxy survey indicates that the standard extrapolated QSO
luminosity function may slightly over-predict the QSO contribution to
J_{\nu}(912) at z ~ 3. We briefly discuss the implications of a
galaxy-dominated UV background at high redshifts.
