  The Infrared Array Camera (IRAC) on the Spitzer Space Telescope currently
offers the greatest potential for high-precision astrometry of faint mid-IR
sources across arcminute-scale fields, which would be especially valuable for
measuring parallaxes of cold brown dwarfs in the solar neighborhood and proper
motions of obscured members of nearby star-forming regions. To more fully
realize IRAC's astrometric capabilities, we have sought to minimize the largest
sources of uncertainty in astrometry with its 3.6 and 4.5 $\mu$m bands. By
comparing different routines that estimate stellar positions, we have found
that Point Response Function (PRF) fitting with the Spitzer Science Center's
Astronomical Point Source Extractor produces both the smallest systematic
errors from varying intra-pixel sensitivity and the greatest precision in
measurements of positions. In addition, self-calibration has been used to
derive new 7$^{\rm th}$ and 8$^{\rm th}$ order distortion corrections for the
3.6 and 4.5 $\mu$m arrays of IRAC, respectively. These corrections are suitable
for data throughout the mission of Spitzer when a time-dependent scale factor
is applied to the corrections. To illustrate the astrometric accuracy that can
be achieved by combining PRF fitting with our new distortion corrections, we
have applied them to archival data for a nearby star-forming region, arriving
at total astrometric errors of $\sim$20 and 70 mas at signal to noise ratios of
100 and 10, respectively.
