We consider an extension of the MSSM wherein anomaly mediation is the source of supersymmetry breaking, and the tachyonic slepton problem is solved by a gauged U(1) symmetry, which is broken at high energies in a manner preserving supersymmetry, thereby also facilitating the see-saw mechanism for neutrino masses and a natural source for the Higgs mu-term. We show that these favourable outcomes can occur both in the presence and the absence of a large Fayet-Iliopoulos (FI) D-term associated with the new U(1). We explore the cosmological consequences of the model, showing that it naturally produces a period of hybrid inflation, terminating in the production of cosmic strings. In spite of the presence of a U(1) (even with an FI term), inflation is effected by the F-term, with a D-flat tree potential (the FI term, if present, being cancelled by non-zero squark and slepton fields). Calculating the 1-loop corrections to the inflaton potential, we estimate the constraints on the parameters of the model from Cosmic Microwave Background data. We will see that a consequence of these constraints is that the Higgs mu-term necessarily small. We briefly discuss the mechanisms for baryogenesis via conventional leptogenesis, the out-of-equilibrium production of neutrinos from the cosmic strings, or the Affleck-Dine mechanism. Cosmic string decays also boost the relic density of dark matter above the low value normally obtained in AMSB scenarios.