  An analysis of the neutron EDM and of the electron EDM in minimal N=1
supergravity unification with two CP violating phases is given. For the neutron
the analysis includes the complete one loop gluino, chargino, and neutralino
exchange diagrams for the electric dipole and the chromoelectric dipole
operators, and also the contribution of the purely gluonic dimension six
operator. It is shown that there exist significant regions in the six
dimensional parameter space of the model where cancellations between the gluino
and the chargino exchanges reduce the electric and the chromoelectric
contributions, and further cancellations among the electric, the
chromoelectric, and the purely gluonic parts lead to a dramatic lowering of the
neutron EDM sometimes below the electron EDM value. This phenomenon gives a new
mechanism, i.e., that of internal cancellations, for the suppression of the
neutron EDM in supersymmetric theories. The cancellation mechanism can
significantly reduce the severe fine tuning problem associated with CP
violating phases in SUSY and SUGRA unified models.
