  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.

Astrophysics