  We present a qualitatively new mechanism for dynamical spontaneous breakdown
of supersymmetry. Specifically, we construct a modified formulation of standard
minimal N=1 supergravity. The modification is based on an idea worked out in
detail in previous publications by some of us, where we proposed a new
formulation of (non-supersymmetric) gravity theories employing an alternative
volume form (volume element, or generally-covariant integration measure) in the
pertinent Lagrangian action, defined in terms of auxiliary (pure-gauge) fields
instead of the standard Riemannian metric volume form. Invariance under
supersymmetry of the new modified N=1 supergravity action is preserved due to
the addition of an appropriate compensating antisymmetric tensor gauge field.
This new formalism naturally triggers the appearance of a dynamically generated
cosmological constant as an arbitrary integration constant which signifies a
spontaneous (dynamical) breaking of supersymmetry. Furthermore, applying the
same formalism to anti-de Sitter supergravity allows us to appropriately choose
the above mentioned arbitrary integration constant so as to obtain
simultaneously a very small positive effective observable cosmological constant
as well as a large physical gravitino mass as required by modern cosmological
scenarios for slowly expanding universe of today.
