The evolution of the quasar luminosity function (QLF) is one of the basic cosmological measures providing insight into structure formation and mass assembly in the universe. We have conducted a spectroscopic survey to find faint quasars (-26.0 < M{sub 1450} < -22.0) at redshifts z = 3.8-5.2 in order to measure the faint end of the QLF at these early times. Using available optical imaging data from portions of the NOAO Deep Wide-Field Survey and the Deep Lens Survey, we have color-selected quasar candidates in a total area of 3.76 deg{sup 2}. Thirty candidates have R <= 23 mag. We conducted spectroscopic follow-up for 28 of our candidates and found 23 QSOs, 21 of which are reported here for the first time, in the 3.74 < z < 5.06 redshift range. We estimate our survey completeness through detailed Monte Carlo simulations and derive the first measurement of the density of quasars in this magnitude and redshift interval. We find that the binned luminosity function (LF) is somewhat affected by the K-correction used to compute the rest-frame absolute magnitude at 1450 A. Considering only our R <= 23 sample, the best-fit single power law (PHI {proportional_to} L {sup beta}) gives a faint-end slope beta = -1.6 +- 0.2. If we consider our larger, but highly incomplete sample going 1 mag fainter, we measure a steeper faint-end slope -2 < beta < -2.5. In all cases, we consistently find faint-end slopes that are steeper than expected based on measurements at z {approx} 3. We combine our sample with bright quasars from the Sloan Digital Sky Survey to derive parameters for a double-power-law LF. Our best fit finds a bright-end slope, alpha = -2.4 +- 0.2, and faint-end slope, beta = -2.3 +- 0.2, without a well-constrained break luminosity. This is effectively a single power law, with beta = -2.7 +- 0.1. We use these results to place limits on the amount of ultraviolet radiation produced by quasars and find that quasars are able to ionize the intergalactic medium at these redshifts.