We show that in a minimal extension of the MSSM by means of an extra U(1)
gauge group, the negative mass-squared problem characteristic of the Anomaly
Mediated Supersymmetry Breaking scenario is naturally solved by means of
Fayet-Iliopoulos D-terms. We derive a set of sum rules for the sparticle masses
which are consequences of the resulting framework.Comment: 11 pages, including 1 figure. Plain TeX. Uses Harvmac and epsf. Final
version to appear in Phys. Lett. B; some minor improvement