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The superstable Weakly Imperfect Bose Gas {(WIBG)} was originally derived tosolve the inconsistency of the Bogoliubov theory of superfluidity. Itsgrand-canonical thermodynamics was recently solved but not at {point of} the{(first order)} phase transition. This paper proposes to close this gap byusing the large deviations formalism and in particular the analysis of the Kacdistribution function. It turns out that, as a function of the chemicalpotential, the discontinuity of the Bose condensate density at the phasetransition {point} disappears as a function of the particle density. Indeed,the Bose condensate continuously starts at the first critical particle densityand progressively grows but the free-energy per particle stays constant untilthe second critical density is reached. At higher particle densities, the Bosecondensate density as well as the free-energy per particle both increase{monotonously}.

Large Deviations in the Superstable Weakly Imperfect Bose Gas