  We demonstrate that a certain class of low scale supersymmetric
``Nelson-Barr'' type models can solve the strong and supersymmetric CP problems
while at the same time generating sufficient weak CP violation in the
$K^{0}-\bar{K}^{0}$ system. In order to prevent one-loop corrections to
$\bar{\theta}$ which violate bounds coming from the neutron electric dipole
moment (EDM), one needs a scheme for the soft supersymmetry breaking parameters
which can naturally give sufficient squark degeneracies and proportionality of
trilinear soft supersymmetry-breaking parameters to Yukawa couplings. We show
that a gauge-mediated supersymmetry breaking sector can provide the needed
degeneracy and proportionality, though that proves to be a problem for generic
Nelson-Barr models. The workable model we consider here has the Nelson-Barr
mass texture enforced by a gauge symmetry; one also expects a new U(1) gauge
superfield with mass in the TeV range. The resulting model is predictive. We
predict a measureable neutron EDM and the existence of extra vector-like quark
superfields which can be discovered at the LHC. Because the $3\times 3$
Cabbibo-Kobayashi-Maskawa matrix is approximately real, the model also predicts
a flat unitarity triangle and the absence of substantial CP violation in the
$B$ system at future $B$ factories. We discuss the general issues pertaining to
the construction of such a workable model and how they lead to the successful
strategy. A detailed renormalization group study is then used to establish the
feasibility of the model considered.
