Diffuser-augmented wind turbines are known for the potential improvement in power extraction in comparison with open wind turbines. Despite the large number of research works dealing with this subject and unlike the open rotor case, an optimum ducted rotor model is still missing. Since the Joukowsky (free-vortex) optimum rotor exhibits the best power coefficient in the open configuration, this paper presents a newly developed ring-vortex free-wake approach for the performance evaluation of an optimum Joukowsky rotor enclosed in a duct of general shape. The method, which is extensively verified, relies on the exact solution of the steady, incompressible, inviscid and axisymmetric flow, and it naturally takes into account the wake divergence and rotation. The procedure is used to obtain, for the first time, the maximum-power-coefficient/tip-speed-ratio characteristic curve for a diffuser augmented wind turbine. The proposed ducted rotor beats the Betz limit by 14.5% when the power coefficient is referred to the device frontal (exit) area. Additionally, the device experiences a slower decrease in performance with the reduction of the tip-speed ratio, thus extending the design range of ducted rotors in comparison with the open ones. Finally, taking into account the mutual influence of the disk and duct, a new rotor design strategy, capable to evaluate the optimum distribution of the chord and pitch-angle along the blade span, is also proposed. A complete design exercise is carried out and the rotor geometry is obtained for three different values of the nominal tip speed ratio. The paper also proves that a two-dimensional design procedure, which strongly couples the duct and the rotor induced flow, is mandatory to properly evaluate the optimum rotor geometry.