This paper presents a new scheme for a discrete adaptive observer for single-input single-output linear plants. In this scheme, the observer is constructed in parametrized form, and the adaptive law for parameter adjustment is given by a set of recursive equations which are derived on the basis of an exponentially weighted least-squares method. The proposed adaptive observer converges rapidly and also is able to track time-varying plant parameters quickly. Hence, the technique, can be well used as an identifier for indirect adaptive control systems. Its effectiveness has been demonstrated by computer simulations carried out for a second-order plant.