  We introduce an $N=1$ supergravity model based on the gauged shift symmetry
of a single chiral multiplet, which can be identified with the string dilaton
or a compactification modulus. The model allows for a tunably small and
positive value of the cosmological constant. The gravitino mass parameter and
the dilaton Vacuum Expectation Value are separately tunable.
  In a second part we analyze the quantum consistency of these models. Recent
work on anomalies in supergravity theories with Fayet-Iliopoulos terms was
extended, such that their results can be interpreted from a field-theoretic
point of view. We show that for certain values of the parameters the anomaly
cancellation conditions are inconsistent with a TeV gravitino mass.
  In the third part the above model is used as a hidden sector for
supersymmetry breaking. In its minimal version, the model leads to tachyonic
scalar soft masses. This problem can however be circumvented by the
introduction of an extra Pol\'onyi-like hidden sector field, or by allowing for
non-canonical kinetic terms for the Standard Model fields, while maintaining
the desirable features of the model. The resulting low energy spectrum consists
of very light neutralinos, charginos and gluinos, while the squarks remain
heavy, with the exception of the stop squark which can be as light as 2 TeV.
  Finally, we discus possibility that the shift symmetry is identified with
known global symmetries of the MSSM. The particular cases where this global
symmetry is Baryon minus Lepton number ($B-L$), or $3B-L$, which contain the
known R-parity or matter parity of the MSSM, are analyzed in great detail. It
is shown that the phenomenology is similar to the above case where the MSSM
fields are inert, with the exception of the stop squark, which can be as light
as $1.5$ TeV.
