  Supergravity models are constructed in which the effective low energy theory
contains only ``super-soft'' explicit supersymmetry breaking: masses of the
scalars and pseudoscalars within a multiplet are split in opposite directions.
With this form of supersymmetry breaking the radiative corrections of the
matter sector to the vacuum energy are bounded by ${\cal O} (M^4_{Susy})$ to
all orders in perturbation theory, and we require $Str M^2 = 0$ including the
hidden sector. The models are based on K\"ahler potentials obtained in recent
orbifold compactifications, and we describe the construction of realistic
theories.
