This paper completely solves the optimal Weighted Least Mean Square (WLMS) design problem using sums of separable terms. For any fixed number of separable terms (less than or equal to the rank of the unconstrained solution), the problem is solved as a sequence of separable filter approximations. An efficient computational algorithm based on necessary conditions is presented. The procedure allows a high degree of flexibility in the choice of filter orders and the number of separable terms, but it may converge to a local minimum. An improved approximation can be obtained by computing more terms than required and then performing a truncation of the coefficient matrix using a singular value analysis. A significant computational advantage is that the procedure requires neither the solution of the unconstrained WLMS problem nor the singular value analysis of the ideal filter.