L. Bertini, E.N.M. Cirillo, E. Olivieri Renormalization Group in the uniqueness region: weak Gibbsianity and convergence. (897K, Postscript) ABSTRACT. We analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature. %We discuss the %Gibbs property of the renormalized measure and the convergence of %renormalized potential under iteration of the map. We prove weak Gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense. Since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system: the correlation length or the critical length for metastability, or both. Thus, to perturbatively treat the problem we have to use a scale--adapted expansion. Moreover, such a model below the critical temperature resembles a disordered system in presence of Griffiths' singularity. Then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached.