  The All-Sky Monitor (ASM) on the Rossi X-ray Timing Explorer (RXTE) has been
used to localize thirteen confirmed X-ray counterparts to Gamma-Ray Bursts
(GRBs) detected over three years of operation. We quantify the errors in ASM
localizations of brief transient sources by using observations of persistent
sources with well-known locations. We apply the results of this analysis to
obtain accurate error boxes with reliable confidence levels for the thirteen
GRBs. In six of these thirteen cases, multiple detections by the ASM allow the
positions to be localized to a diamond of order ~15' x 3'. In five further
cases, the Interplanetary Network (IPN) constrains the usually ~3 deg. x 3'
(full-width) ASM error box to an area of a few tens of square arcminutes. This
work adds eleven burst localizations to the list of ~60 well-localized GRBs.
