We consider the problem of adaptive filtering for ergodic Ornstein-Uhlenbeck process observed in the presence of white Gaussian noise. The asymptotic normality and convergence of moments of the EMM, MLE, and BE are established. Preliminary estimators of the unknown parameters are constructed by the observations on the learning interval with the help of the method of moments. These estimators are then used to define the One-step MLE-process, and substituting of this estimator process into the equations of Kalman-Bucy filtering yields the recursive adaptive filter. The properties of all estimators and the approximation error of the conditional expectation of the Ornstein-Uhlenbeck process are described in the asymptotics of large samples. The asymptotic efficiency of the proposed adaptive filter is discussed too.