  We consider theories with one or more compact dimensions with size r > 1/M,
where M is the fundamental Planck scale, with the visible and hidden sectors
localized on spatially separated "3-branes." We show that a bulk U(1) gauge
field spontaneously broken on the hidden-sector 3-brane is an attractive
candidate for the messenger of supersymmetry breaking. In this scenario scalar
mass-squared terms are proportional to U(1) charges, and therefore naturally
conserve flavor. Arbitrary flavor violation at the Planck scale gives rise to
exponentially suppressed flavor violation at low energies. Gaugino masses can
be generated if the standard gauge fields propagate in the bulk; \mu and B\mu
terms can be generated by the Giudice-Masiero or by the VEV of a singlet in the
visible sector. The latter case naturally solves the SUSY CP problem. Realistic
phenomenology can be obtained either if all microscopic parameters are order
one in units of M, or if the theory is strongly coupled at the scale M. (For
the latter case, we estimate parameters by extending "naive dimensional
analysis" to higher-dimension theories with branes.) In either case, the only
unexplained hierarchy is the "large" size of the extra dimensions in
fundamental units, which need only be an order of magnitude. All soft masses
are naturally within an order of magnitude of m_{3/2}, and trilinear scalar
couplings are negligible. Squark and slepton masses can naturally unify even in
the absence of grand unification.
