This paper is concerned with the output feedback exponential stabilization for a one-dimensional wave equation with in-domain feedback/recirculation of a boundary velocity with a spatially constant coefficient, which is first studied in [IEEE Trans. Autom. Control, vol. 62, no. 9, pp. 4760-4767, Sep. 2017]. When there are no boundary internal uncertainty and external disturbance, it is shown that by using one displacement measurement only, the output feedback makes the closed-loop system exponentially stable, which essentially improves the result of [IEEE Trans. Autom. Control, vol. 62, no. 9, pp. 4760-4767, Sep. 2017]. When there are boundary internal uncertainty and external disturbance, using two displacement measurements only, we present an observer-based output feedback law that contains an infinite-dimensional disturbance estimator used to reject the boundary internal uncertainty and external disturbance. The resulting closed-loop system is shown to be exponentially stable and the state of all subsystem involved are uniformly stable. The Backstepping method for infinite dimensional system and active disturbance rejection control method play important roles in the design.