Covariance control theory provides a parametrization of all controllers that assign a given state covariance matrix to the closed-loop system. Such controllers are not unique. In this note we obtain the analytic expressions for the covariance controllers that minimize the required control effort. These are solutions to a minimum-effort/guaranteed-performance design problem, in the sense that the required control effort is minimized subject to performance constraints guaranteed by a specified state covariance matrix assignment. In a different interpretation of the results, the solutions to a minimum-effort controller redesign problem are obtained. Both continuous-time and discrete-time results are presented.