AbstractWe consider the generalization of the Pólya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under theuniform ergodicityassumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with abranching Markov chainon a weightedrandom recursive treeas described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.