  We present a catalog of 100,563 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 22,737 sources reveals that 22,191
(97.6%) are quasars; accounting for the magnitude dependence of this
efficiency, we estimate that 95,502 (95.0%) of the objects in the catalog are
quasars. Such a high efficiency is unprecedented in broad-band 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 4-D color distribution of stars and spectroscopically
confirmed quasars in the SDSS. Specifically, we use a non-parametric 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. Future work
needed to extend the this selection algorithm to larger redshifts, fainter
magnitudes, and resolved sources is discussed. Finally, we examine some science
applications of the catalog, particularly a tentative quasar number counts
distribution covering the largest range in magnitude (14.2<g<21.0) ever made
within the framework of a single quasar survey.
