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Navigating, guiding, and controlling autonomous underwater vehicles (AUVs) are challenging and difficult tasks compared to the autonomous surface-level operations. Controlling the motion of such vehicles require the estimation of unknown hydrodynamic forces and moments and disturbances acting on these vehicles in the underwater environment. In this paper, a one-layer neural-network (NN) controller with preprocessed input signals is designed to control the vehicle track along a desired trajectory, which is specified in terms of desired position and attitude. In the absence of unknown disturbances and modeling errors, it is shown that the tracking error system is asymptotically stable. In the presence of any bounded ocean currents or wave disturbances, the uniformly ultimately boundedness of the tracking error and NN weight estimates are given. The NN does not require an initial offline training phase and weight initialization is straightforward. Simulation results are shown by using a scaled version of the Naval Post-Graduate School's AUV. Results indicate the superior performance of the NN controller over conventional controllers. Providing offline NN training may improve the transient performance.