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In this study, a single neural network (SNN)-based adaptive control design method is developed for a class of uncertain non-affine pure-feedback non-linear systems. Different from existing methods, all unknown parts at intermediate steps are passed down, and only an SNN is used to approximate the lumped unknown function of the system at the last step of controller design. By this approach, the designed controller consisting of an actual control law and an adaptive law can be given directly, and the complexity growing problem inherent in conventional methods can be completely eliminated. Stability analysis shows that all the closed-loop system signals are uniformly ultimately bounded, and the steady-state tracking error can be made arbitrarily small by appropriately choosing control parameters. Simulation results demonstrate the effectiveness of the proposed approach.