  We use a new technique to search for faint red stars in the Hubble Deep Field
(HDF) imaged by the Wide Field Camera (WFC2) on the {\it Hubble Space
Telescope}. We construct a densely sampled stellar light profile from a set of
undersampled images of bright stars in a low-latitude field. Comparison of this
stellar light profile to densely sampled profiles of individual objects in the
HDF constructed from multiple undersampled dithers allows us to distinguish
unambiguously between stars and galaxies to $I=26.3$. We find no stars with
$V-I>1.8$ in the outer 90\% of the volume probed. This result places strong and
general constraints on the $I$ band luminosity of the constituents of the
Galactic dark halo: $$M_I > 15.9 + {5\over 3}\log\biggl(f{0.5 M_\odot\over
M}\biggr)\qquad (V-I>1.8),$$ where $M$ is the mass of the objects and $f$ is
their density as a fraction of the local halo density, taken to be
$\rho_0=9\times 10^{-3}\,M_\odot\,\pc^{-3}$. If the halo is made of white
dwarfs, this limit implies that these objects have $M_V\gsim 18.4$ and
$V-I\gsim 2.5$. That is, they are $\gsim 2$ magnitudes fainter than the end of
the disk white dwarf sequence. Faint red dwarfs account for $<1\%$ of the
Galactic dark halo for $M_I<14$, and $<6\%$ for $M_I<15$ at the 95\% confidence
level. The density of inter-galactic Local Group stars is at least a factor
3000 smaller than the density of local Population II stars.
