In this paper an adaptive observer for a class of uncertain nonlinear systems is proposed. Based on linearly parameterized neural networks, Lyapunov argument, and an adaptive bounding technique, the proposed scheme ensures zero observer error convergence, asymptotically, even in the presence of approximation error and disturbances, whereas the others error signals remain bounded. In addition, the proposed scheme does not rely on any Riccati equation solution and it does not suffer from chattering.