This paper addresses the problem of asymptotic trajectory tracking with prescribed transient behavior for a fully actuated marine surface vessel subject to unmodeled dynamics and external disturbances. A prescribed transient performance constraint is introduced to specify the transient behavior of the output tracking errors. Neural network (NN) approximation and adaptive control techniques are employed to compensate for the unmodeled dynamics. In general, the NN approximation errors always exist in the NN-based adaptive control systems, and typically result in uniformly ultimate boundedness, where the tracking errors cannot be stabilized asymptotically. In order to obtain an asymptotic tracking performance, a robust integral of the sign of the error (RISE) feedback term is introduced in feedback control design to compensate for the NN approximation errors and external disturbances. Consequently, RISE-based adaptive NN controller with compensations of unmodeled dynamics and external disturbances is proposed such that an asymptotic tracking performance is achieved with guaranteeing prescribed transient behavior. The performance of the proposed tracking controller is shown by simulation results.