  We derive an effective Hamiltonian for ${\rm Ga}_{1-x}{\rm Mn}_x {\rm As}$ in
the dilute limit, where ${\rm Ga}_{1-x}{\rm Mn}_x {\rm As}$ can be described in
terms of spin $F=3/2$ polarons hopping between the {\rm Mn} sites and coupled
to the local {\rm Mn} spins. We determine the parameters of our model from
microscopic calculations. Our approach treats the large Coulomb interaction in
a non-perturbative way, captures the effects of spin-orbit coupling and
disorder, and is appropriate for other p-doped magnetic semiconductors. Our
model applies to uncompensated {\rm Mn} concentrations up to $x \sim 0.03$.
