We present improved and highly accurate International Celestial Reference System (ICRS) equatorial positions for 1,268,732 stars in 16 astrometric calibration regions located around the celestial equator. Each region is about 7.6dx3.2d in area (~24.3 deg2), contains a large number of stars (18,767 to 263,810), has stellar densities ranging from 765 to 10,772 stars deg-2, covers a wide range in magnitude (9.5<R<17.8, or equivalently 10.0<V<18.3), and is complete to magnitude R~17.2 (V~17.7). All of the observations were taken in 1994-1998 (mean epoch ~J1996.0) with the Flagstaff Astrometric Scanning Transit Telescope. Furthermore, each region was observed many times with overlapped CCD strip scans, and these data were reduced to star positions using differential reductions. The reference-star positions were taken from the ACT catalog of accurate star positions and proper motions. The star positions presented herein are typically accurate to +/-26 mas (+/-12 mas precision) in both right ascension and declination, except for the faintest stars, whose errors are about 2 times larger. Similarly, magnitudes were determined for the astrometric calibration stars using standard photometric reductions, and accuracies of +/-18 mmag (+/-10 mmag precisions) were achieved (about +/-46 mmag for the faintest stars). There is little variation among the individual astrometric calibration regions in terms of the positional and photometric accuracies achieved. In addition, the positions and magnitudes presented in this paper supersede those given in 1997 by Stone, insofar as they are considerably more accurate, are rigorously placed in the ICRS, include many more stars in the low Galactic regions, and have had accurate proper motions computed for them as well. The proper motions are accurate to about +/-6 mas yr-1 and should maintain the accuracy of the positions for many years to come. Concerning possible applications, the positions and magnitudes of the stars in the astrometric calibration regions can be used to calibrate single and mosaicked CCD detectors, as well as to optimize techniques for reducing star measures on astrographic plates to star positions and magnitudes. As a consequence, the positions, proper motions, and magnitudes put forth in this paper are made available to interested users.