We present a catalog of 100563 unresolved, UV-excess (UVX) quasar candidates to g=21 from 2099 deg^2^ of the Sloan Digital Sky Survey (SDSS) Data Release One (DR1) imaging data. Existing spectra of 22737 sources reveals that 22191 (97.6%) are quasars; accounting for the magnitude dependence of this efficiency, we estimate that 95502 (95.0%) of the objects in the catalog are quasars. Such a high efficiency is unprecedented in broadband surveys of quasars. This "proof-of-concept" sample is designed to be maximally efficient, but still has 94.7% completeness to unresolved, g~<19.5, UVX quasars from the DR1 quasar catalog. This efficient and complete selection is the result of our application of a probability density type analysis to training sets that describe the four-dimensional color distribution of stars and spectroscopically confirmed quasars in the SDSS. Specifically, we use a nonparametric Bayesian classification, based on kernel density estimation, to parameterize the color distribution of astronomical sources - allowing for fast and robust classification. We further supplement the catalog by providing photometric redshifts and matches to FIRST/VLA, ROSAT, and USNO-B sources.