This paper considers a stable state observer for bilinear systems. The design of the proposed observer is based on Lyapunov stability and the norm of the estimation error decays to zero exponentially irrespective of the input, although the estimation error itself depends on the input. Sufficient conditions are given for the existence of a stable observer. These conditions make it possible to design a stable observer for a class of bilinear systems, which has no observer by the other method.