Several neural network (NN) models have been applied successfully for modeling complex nonlinear dynamical systems. However, the stable adaptive state estimation of an unknown general nonlinear system from its input and output measurements is an unresolved problem. This paper addresses the nonlinear adaptive observer design for unknown general nonlinear systems. Only mild assumptions on the system are imposed: output equation is at least C(1) and existence and uniqueness of solution for the state equation. The proposed observer uses linearly parameterized neural networks (LPNNs) whose weights are adaptively adjusted, and Lyapunov theory is used in order to guarantee stability for state estimation and NN weight errors. No strictly positive real (SPR) assumption on the output error equation is required for the construction of the proposed observer.