Exact results are provided for the ground-state properties of an impurity spin coupled to a conducting band of strongly correlated electrons described by a t-J model. The ground states are proved to be unique, apart from the spin rotational degeneracy, and the ground-state spins are found in the following cases: (1) J=0, one-hole hopping on a lattice in dimensions d2 and (2) J0, arbitrary number of holes hopping on a one-dimensional lattice, i.e., a Luttinger liquid interacting with a localized impurity spin. © 1995 The American Physical Society.