We present a measurement of the Type I quasar luminosity function at z = 5 using a large sample of spectroscopically confirmed quasars selected from optical imaging data. We measure the bright end (M 1450 <–26) with Sloan Digital Sky Survey (SDSS) data covering ~6000 deg2, then extend to lower luminosities (M 1450 <–24) with newly discovered, faint z ~ 5 quasars selected from 235 deg2 of deep, coadded imaging in the SDSS Stripe 82 region (the celestial equator in the Southern Galactic Cap). The faint sample includes 14 quasars with spectra obtained as ancillary science targets in the SDSS-III Baryon Oscillation Spectroscopic Survey, and 59 quasars observed at the MMT and Magellan telescopes. We construct a well-defined sample of 4.7 < z < 5.1 quasars that is highly complete, with 73 spectroscopic identifications out of 92 candidates. Our color selection method is also highly efficient: of the 73 spectra obtained, 71 are high-redshift quasars. These observations reach below the break in the luminosity function (M_{1450}^*\approx -27). The bright-end slope is steep (β <~ –4), with a constraint of β <–3.1 at 95% confidence. The break luminosity appears to evolve strongly at high redshift, providing an explanation for the flattening of the bright-end slope reported previously. We find a factor of ~2 greater decrease in the number density of luminous quasars (M 1450 <–26) from z = 5 to z = 6 than from z = 4 to z = 5, suggesting a more rapid decline in quasar activity at high redshift than found in previous surveys. Our model for the quasar luminosity function predicts that quasars generate ~30% of the ionizing photons required to keep hydrogen in the universe ionized at z = 5. Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. This paper also includes data gathered with the 6.5 m Magellan telescopes located at Las Campanas Observatory, Chile.