We present a new measurement of the optical quasar luminosity function (QLF), using data from the Sloan Digital Sky Survey-III: Baryon Oscillation Spectroscopic Survey (SDSS-III: BOSS). From the SDSS-III Data Release Nine, a uniform sample of 22,301 i <~ 21.8 quasars are selected over an area of 2236 deg2, with confirmed spectroscopic redshifts between 2.2 < z < 3.5, filling in a key part of the luminosity-redshift plane for optical quasar studies. The completeness of the survey is derived through simulated quasar photometry, and this completeness estimate is checked using a sample of quasars selected by their photometric variability within the BOSS footprint. We investigate the level of systematics associated with our quasar sample using the simulations, in the process generating color-redshift relations and a new quasar K-correction. We probe the faint end of the QLF to Mi (z = 2.2) ≈ –24.5 and see a clear break in the QLF at all redshifts up to z = 3.5. A log-linear relation (in log Φ* – M*) for a luminosity evolution and density evolution model is found to adequately describe our data within the range 2.2 < z < 3.5; across this interval the break luminosity increases by a factor of ~2.6 while Φ* declines by a factor of ~8. At z <~ 2.2 our data are reasonably well fit by a pure luminosity evolution model, and only a weak signature of "AGN downsizing" is seen, in line with recent studies of the hard X-ray luminosity function. We compare our measured QLF to a number of theoretical models and find that models making a variety of assumptions about quasar triggering and halo occupation can fit our data over a wide range of redshifts and luminosities.