  It has often been suggested that the cosmic positron excess observed by the
HEAT experiment could be the consequence of supersymmetric dark matter
annihilating in the galactic halo. Although it is well known that evenly
distributed dark matter cannot account for the observed excess, if substantial
amounts of local dark matter substructure are present, the positron flux would
be enhanced, perhaps to the observed magnitude. In this paper, we attempt to
identify the nature of the substructure required to match the HEAT data,
including the location, size and density of any local dark matter clump(s).
Additionally, we attempt to assess the probability of such substructure being
present. We find that if the current density of neutralino dark matter is the
result of thermal production, very unlikely ($\sim 10^{-4}$ or less) conditions
must be present in local substructure to account for the observed excess.
