  We construct phenomenologically viable supersymmetric models where CP is an
approximate symmetry. The full high energy theory has exact CP and horizontal
symmetries that are spontaneously broken with a naturally induced hierarchy of
scales, $\Lambda_{CP}\ll\Lambda_H$. Consequently, the effective low energy
theory, that is the supersymmetric Standard Model, has CP broken explicitly but
by a small parameter. The $\epsilon_K$ parameter is accounted for by
supersymmetric contributions. The predictions for other CP violating
observables are very different from the Standard Model. In particular, CP
violating effects in neutral B decays into final CP eigenstates such as
$B\ra\psi K_S$ and in $K\ra\pi\nu\bar\nu$ decays are very small.
