This article presents and discusses a method for measuring the proper motions of the Galactic dwarf spheroidal galaxies using images taken with the Hubble Space Telescope. The method involves fitting an effective point spread function to the image of a star or quasi-stellar object to determine its centroid with an accuracy of about 0.005 pixel (0.25 milliarcseconds) -- an accuracy sufficient to measure the proper motion of a dwarf spheroidal galaxy using images separated by just a few years. The data consist of images, dithered to reduce the effects of undersampling, taken at multiple epochs with the Space Telescope Imaging Spectrograph or the Wide Field Planetary Camera. The science fields are in the directions of the Carina, Fornax, Sculptor, and Ursa Minor dwarf spheroidal galaxies and each has at least one quasi-stellar object whose identity has been established by other studies. The rate of change with time of the centroids of the stars of the dwarf spheroidal with respect to the centroid of the quasi-stellar object is the proper motion. Four independent preliminary measurements of the proper motion of Fornax for three fields agree within their uncertainties. The weighted average of these measurements is mu_alpha = 49 +- 13 milliarcseconds/century and mu_delta = -59 +- 13 milliarcseconds/century. The Galactocentric velocity derived from the proper motion implies that Fornax is near perigalacticon, may not be bound to the Milky Way, and is not a member of any of the proposed streams of galaxies and globular clusters in the Galactic halo. If Fornax is bound, the Milky Way must have a mass of at least (1.6 +- 0.8) x 10^{12} solar masses.
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