In this paper, a nonlinear controller based on adaptive sliding-mode method which has a sliding surface vector including new boundizing function is proposed and applied to a two-wheeled welding mobile robot (WMR). This controller makes the welding point of WMR achieve tracking a reference point which is moving on a smooth curved welding path with a desired constant velocity. The mobile robot is considered in view of a kinematic model and a dynamic model in Cartesian coordinates. The proposed controller can overcome uncertainties and external disturbances by adaptive sliding-mode technique. To design the controller, the tracking error vector is defined, and then the sliding surface vector including new boundizing function and the adaptation laws are chosen to guarantee that the error vector converges to zero asymptotically. The stability of the dynamic system is shown through the Lyapunov method. In addition, a simple way of measuring the errors by potentiometers is introduced. The simulations and experimental results are shown to prove the effectiveness of the proposed controller.