The Flagstaff Astrometric Scanning Transit Telescope was used to scan the sky in search of occultation candidates through which the Centaur 10199 Chariklo (formerly 1997 CU_26_) will pass in 1999-2005. Positions for 96,397 stars (magnitudes 7.5<V<17.3) were determined using differential reductions to the ACT catalog of star positions and proper motions, and accuracies of +/-30mas were achieved for well-exposed images. The ephemeris for 10199 Chariklo was improved by including the new positions presented in this paper and, second, by correcting older positions taken from the literature for systematic errors. During this time period, 10199 Chariklo passes within 2" of 117 of these stars, which are identified as occultation candidates in this paper. Among these, 28 candidates have magnitudes V<15.0, making them the best choices for observing programs using portable telescopes. Circumstances for each occultation are given. Because the angular radius of 10199 Chariklo is only about 15mas, the astrometry provided in this paper is only accurate enough for identifying possible occultation events and is not sufficient for predicting individual shadow paths across the surface of Earth, although representative cases are given. Last-minute astrometry with a large-aperture telescope will be needed to refine each prediction.