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 theeffectivePSF (ePSF) 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 theHubble Space TelescopePSF. 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 \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackageOT2,OT1{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $\sqrt{N}$ \end{document} .