A general computationally efficient procedure is developed to synthesize the primary illumination of a focusing reflector antenna and obtain the best fit to a given far-field amplitude pattern. The so-called power synthesis approach is considered, wherein the design specifications concern only the amplitude of the required far field. The mathematical framework of the synthesis problem is discussed, revealing the difficulties related to its ill-posedness and to the nonuniqueness of the solutions. The power-synthesis procedure is split into two steps: synthesis of the current induced on the reflector surface and subsequent determination of the primary field. The current synthesis is then considered, and a very efficient algorithm is developed. For determining the primary far-field, the analytic properties of the synthesized current are used to continue it outside the original domain of definition. The cancellation principle is then used to determine the required primary pattern.