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Let (X _jk ) _jk≥1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ ². Let M be the n × n random Markov matrix with i.i.d. rows defined by . In particular, when X ₁₁ follows an exponential law, the random matrix M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Let λ₁, . . . , λ _n be the eigenvalues of i.e. the roots in of its characteristic polynomial. Our main result states that with probability one, the counting probability measure converges weakly as n→∞ to the uniform law on the disk . The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.