  The stellar initial mass function at high redshift is an important defining
property of the first stellar systems to form and may also play a role in
various dark matter problems. We here determine the faint stellar luminosity
function in an apparently dark-matter-dominated external galaxy in which the
stars formed at high redshift. The Ursa Minor dwarf spheroidal galaxy is a
system with a particularly simple stellar population - all of the stars being
old and metal-poor - similar to that of a classical halo globular cluster. A
direct comparison of the faint luminosity functions of the UMi Sph and of
similar metallicity, old globular clusters is equivalent to a comparison of the
initial mass functions and is presented here, based on deep HST WFPC2 and STIS
imaging data. We find that these luminosity functions are indistinguishable,
down to a luminosity corresponding to 0.3 solar masses. Our results show that
the low-mass stellar IMF for stars that formed at very high redshift is
apparently invariant across environments as diverse as those of an extremely
low-surface-brightness, dark-matter-dominated dwarf galaxy and a
dark-matter-free, high-density globular cluster within the Milky Way.
