  We have used HST imaging of the central regions (R<100 arcsec, about 5 core
radii) of the globular cluster 47 Tucanae to derive proper motions and U- and
V-band magnitudes for 14,366 cluster members. We also present a catalogue of
astrometry and F475W photometry for nearly 130,000 stars in a rather larger
central area. These data are made available in their entirety, in the form of
downloadable electronic tables. We use them first to obtain a new estimate for
the position of the cluster center and to define the stellar density profile
into essentially zero radius. We then search in particular for any very
fast-moving stars, such as might be expected to result from very close stellar
encounters. Likely fewer than 0.1% (and no more than about 0.3%) of stars have
total speeds above the nominal central escape velocity in 47 Tuc, and at lower
speeds the velocity distribution is described very well by a regular King
model. Considerations of only the proper-motion velocity dispersion then lead
to a number of results: (1) Blue stragglers in the core of 47 Tuc have a
velocity dispersion lower than that of the cluster giants by a factor of
sqrt{2}. (2) The velocity distribution in the cluster center is essentially
isotropic, as expected. (3) Using a sample of radial velocities for stars in
the core, we estimate the distance to 47 Tuc: D = 4.0 +/- 0.35 kpc. And (4) we
infer a 1-sigma upper limit of M<1000-1500 solar masses for any central,
intermediate-mass black hole. We can neither confirm nor refute the hypothesis
that 47 Tuc might lie on an extension of the M-sigma relation observed for
supermassive black holes in galaxy bulges. [Abridged]
