Power system stabilizers are widely used to generate supplementary control signals for the excitation system in order to damp out the low frequency oscillations. This paper proposes a stable neural network (NN) controller for the stabilization of a single machine infinite bus power system. In the power system control literature, simplified analytical models are used to represent the power system and the controller designs are not based on rigorous stability analysis. This work overcomes the two major problems by using an accurate analytical model for controller development and presents the closed-loop stability analysis. The NN is used to approximate the complex nonlinear power system online and the weights of which can be set to zero to avoid the time consuming offline training process. Magnitude constraint of the activators is modeled as saturation nonlinearities and is included in the Lyapunov stability analysis. Simulation results demonstrate that the proposed design can successfully damp out oscillations. The control algorithms of this work can also be applied to other similar control problems.