We propose a possible new way to resolve the long standing problem of strong supersymmetry breaking coexisting with a small cosmological constant. We consider a scalar component of a minimally coupled N=1 supermultiplet in a general Friedmann-Robertson-Walker (FRW) expanding universe. We argue that a tiny term, proportional to H^2 ~ 10^(-122) in Plank's units, appearing in the field equations due to this expansion will provide both, the small vacuum energy and the heavy mass of the scalar supersymmetric partner. We present a non-perturbative solution for the scalar field with an unusual dual-frequency behavior. This solution has two characteristic mass scales related to the Hubble parameter as H^(1/4) and H^(1/2) measured in Plank's units. Show Partial Abstract