This paper presents a method of simple adaptive control (SAC) using neural networks for a class of nonlinear systems with bounded-input bounded-output (BIBO) and bounded nonlinearity. The control input is given by the sum of the output of the simple adaptive controller and the output of the neural network. The neural network is used to compensate for the nonlinearity of the plant dynamics that is not taken into consideration in the usual SAC. The role of the neural network is to construct a linearized model by minimizing the output error caused by nonlinearities in the control systems. Furthermore, convergence and stability analysis of the proposed method is performed. Finally, the effectiveness of the proposed method is confirmed through computer simulation.