This paper presents a simple setup for ocean current estimation that offers globally asymptotically stable (GAS) error dynamics. In the proposed scenario, the underwater vehicle has only access to limited relative velocity readings, along the longitudinal direction of the vehicle, in addition to position measurements. The observability of the system is analyzed and necessary and sufficient conditions, with physical meaning, are derived, hence useful for motion planning and control. A Kalman filter, with GAS error dynamics, is implemented as a solution to the estimation problem and simulation results are included that illustrate the performance of the proposed solution.