Off-line state identification methods for a sequential machine using a homing sequence or an adaptive homing sequence (AHS) are well-known in the automata theory. There are, however, so far few studies on the subject of the on-line state estimator such as a state observer (SO) which is used in the linear system theory. In this paper, we shall construct such an SO for a Moore machine based on the state identification process by means of AHSs, and discuss the convergence property of the SO.