  The first step toward doing high-precision astrometry is the measurement of
individual stars in individual images, a step that is fraught with dangers when
the images are undersampled. The key to avoiding systematic positional error in
undersampled images is to determine an extremely accurate point-spread function
(PSF). We apply the concept of the {\it effective} PSF, and show that in images
that consist of pixels it is the ePSF, rather than the often-used instrumental
PSF, that embodies the information from which accurate star positions and
magnitudes can be derived. We show how, in a rich star field, one can use the
information from dithered exposures to derive an extremely accurate effective
PSF by iterating between the PSF itself and the star positions that we measure
with it. We also give a simple but effective procedure for representing spatial
variations of the HST PSF. With such attention to the PSF, we find that we are
able to measure the position of a single reasonably bright star in a single
image with a precision of 0.02 pixel (2 mas in WF frames, 1 mas in PC), but
with a systematic accuracy better than 0.002 pixel (0.2 mas in WF, 0.1 mas in
PC), so that multiple observations can reliably be combined to improve the
accuracy by $\surd N$.
