The dynamics of the DNLSE is characterized by peculiar featuresin the region of parameter space above the line at infinite temperature:the deterministic version exhibits multi-breather states,
lasting over astronomical times, while the stochastic (conservative)
evolution yields a coarsening dynamics to an infinite temperature
lattice, with a superimposed giant breather collecting a finite fractionof the total energy. The statistical mechanics of this model can be naturally described and explicitly computed in the microcanonical ensemble and allows us to conclude that the multi breather state, observed in the deterministic evolution, is a genuine equilibrium state at negative temperature. We also show that in this region there is no ensemble equivalence with the grand-canonical ensemble and, moreover, that the infinite temperature line is also the boundary of a first order phase transition between a thermalized (low-energy) phase and a condensed (high-energy) phase. Further details about the presence of a spinodal line, the features of the order parameter and the non-extensivity of the condensed phase will be also discussed.