  We study the ground state properties of a trapped Bose condensate with a
neutral impurity. By varying the strength of the attractive atom-impurity
interactions the degree of localization of the impurity at the trap center can
be controlled. As the impurity becomes more strongly localized the peak
condensate density, which can be monitored experimentally, grows markedly. For
strong enough attraction, the impurity can make the condensate unstable by
strongly deforming the atom density in the neighborhood of the impurity. This
"collapse" can possibly be investigated in bosenova-type experiments.
