We use the deep ground-based optical photometry of the Lyman Break Galaxy (LBG) Survey to derive robust measurements of the faint-end slope ( alpha ) of the UV luminosity function (LF) at redshifts 1.9 <= z <= 3.4. Our sample includes >2000 spectroscopic redshifts and 31000 LBGs in 31 spatially independent fields over a total area of 3261 arcmin2. These data allow us to select galaxies to 0.07L* and 0.10L* at z ~ 2 and z ~ 3, respectively. A maximum-likelihood analysis indicates steep values of alpha (z = 2) = -1.73 ± 0.07 and alpha (z = 3) = -1.73 ± 0.13. This result is robust to luminosity-dependent systematics in the Ly alpha equivalent width and reddening distributions, and is similar to the steep values advocated at z 4, and implies that 93% of the unobscured UV luminosity density at z ~ 2-3 arises from sub-L* galaxies. With a realistic luminosity-dependent reddening distribution, faint to moderately luminous galaxies account for 70% and 25% of the bolometric luminosity density and present-day stellar mass density, respectively, when integrated over 1.9 <= z < 3.4. We find a factor of 8-9 increase in the star-formation rate density between z ~ 6 and z ~ 2, due to both a brightening of L* and an increasing dust correction proceeding to lower redshifts. Combining the UV LF with stellar mass estimates suggests a relatively steep low-mass slope of the stellar mass function at high redshift. The previously observed discrepancy between the integral of the star-formation history and stellar mass density measurements at z ~ 2 may be reconciled by invoking a luminosity-dependent reddening correction to the star-formation history combined with an accounting for the stellar mass contributed by UV-faint galaxies. The steep and relatively constant faint-end slope of the UV LF at z 2 contrasts with the shallower slope inferred locally, suggesting that the evolution in the faint-end slope may be dictated simply by the availability of low-mass halos capable of supporting star formation at z 2.