The problem of unbiased recursive identification of a plant model in closed-loop operation is considered. A particular form of an output error predictor for the closed loop is introduced. This allows one to derive a parameter estimation algorithm for the plant model that is globally asymptotically stable and asymptotically unbiased in the presence of noise. The paper presents a stability analysis in a deterministic environment and a convergence analysis in the stochastic environment. Both require a mild sufficient passivity condition to be satisfied. Simulations and real-time experiments on a flexible transmission illustrate the performances of the proposed algorithm.