  We demonstrate that Dirac neutrino masses in the experimentally preferred
range are generated within supersymmetric gauge extensions of the Standard
Model with a generalized supersymmetry breaking sector. If the usual
superpotential Yukawa couplings are forbidden by the additional gauge symmetry
(such as a U(1)'), effective Dirac mass terms involving the "wrong Higgs" field
can arise either at tree level due to hard supersymmetry breaking fermion
Yukawa couplings, or at one-loop due to nonanalytic or "nonholomorphic" soft
supersymmetry breaking trilinear scalar couplings. As both of these operators
are naturally suppressed in generic models of supersymmetry breaking, the
resulting neutrino masses are naturally in the sub-eV range. The neutrino
magnetic and electric dipole moments resulting from the radiative mechanism
also vanish at one-loop order.
