In certain applications of linear programming, some particular solutions called the weighted centers of the solution set are often desired, giving rise to the need of algorithms capable of locating such centers. In this note, we modify the Mizuno-Todd-Ye predictor-corrector algorithm so that the modified algorithm is guaranteed to converge to the weighted center for given weights. The key idea is to ensure that iterates remain in a sequence of shrinking neighborhoods of the weighted central path. The modified algorithm also possesses polynomiality and superlinear convergence.