A new global output-feedback controller for tracking control of an underactuated surface ship with only surge force and yaw moment available and without measurements of velocities. The reference trajectory to be tracked is allowed to be a curve including a straight-line. A global nonlinear coordinate change is introduced to transform the ship dynamics to a system affine in the ship velocities. A global nonlinear exponential observer is proposed to estimate unmeasured velocities. The controller development is based on the Lyapunov direct method and backstepping technique, and utilizes the passive property of ship dynamics and their interconnected structure.