In this paper, a simple neuron-based adaptive controller for trajectory tracking is developed for nonholonomic mobile robots without velocity measurements. The controller is based on structural knowledge of the dynamics of the robot and the odometric calculation of robot position only. The wheel actuator dynamics is also taken into account. An approximation network approximates a nonlinear function involving robot dynamic parameters so that no knowledge of those parameters is required. The proposed controller is robust not only to structured uncertainty such as mass variation but also to unstructured one such as disturbances. The real-time control of mobile robot is achieved through the online learning of the approximation network. The system stability and the boundness of tracking errors are proved using Lyapunov stability theory. Computer simulations with circular as well as square reference trajectories are presented that confirm the simplicity and effectiveness of the proposed tracking control law. The efficacy of the proposed control law is tested experimentally on a differentially driven mobile robot. Both simulation and experimental results are described in detail.