Remotely Operated Vehicles (ROVs) working close to the sea surface are subjected to wave disturbances that affect their positioning. To treat this problem, we present a control scheme for the dynamic positioning of ROVs under wave disturbances and at low operation depths. The approach is composed of an Augmented Wave Filter (AWF) based on the Extended Kalman Filter (EKF) algorithm and an Adaptive-Model Predictive Control (A-MPC). The filter provides the optimal motion states and wave height estimation to the A-MPC, which calculates the control efforts based on the receding horizon approach. The scheme is tested through numerical simulations on the depth control of the Mandi II-ROV at the diving plane. The results are compared to those of a standard Proportional-Integral-Derivative (PID) controller. To perform realistic simulations, a detailed mathematical model is presented, considering the rigid body-motion, hydrodynamic and hydrostatic forces, thrusters dynamics, and onboard sensor measurements. The simulations are performed for different scenarios, considering the dynamic positioning in still waters and the station-keeping under wave disturbances for significant heights of 1 and 2 m. The controller robustness to parameter variation is assessed by using Monte-Carlo simulation. Results have shown improved performance for the A-MPC in terms of positioning errors, motion oscillations, and drift attenuation in comparison to a PID controller. The Monte-Carlo simulation reveals enhanced robustness to the A-MPC, which is associated with the greater variance for the control forces. Also, the filter performed very well in all the tests, producing an accurate estimate of the wave-induced height and providing an off-set free control.