Abstract : We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on the periodic interval [0,1]. It is based on a model introduced by Bacry and Muzy, and is similar to the logarithmic Random Energy Model studied by Carpentier and Le Doussal, and more recently by Fyodorov and Bouchaud. At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. In particular, this proves a conjecture of Carpentier and Le Doussal that the statistics of the extremes of the log-correlated field behave as those of i.i.d. Gaussian variables and of branching Brownian motion at the level of the Gibbs measure. The proof is based on the computation of the free energy of a perturbation of the model, where a scale-dependent variance is introduced, and on general tools of spin glass theory.