To solve the Euclidean multifacility location problem, W. Miehle extended the Weiszfeld algorithm from one to several facilities. L. M. Ostresh established that the objective function for the iterates of Miehle's algorithm decreases in value, but his results do not give a convergence proof. In this paper we give a new extension of the Weiszfeld algorithm and by generalizing the Kuhn-Ostresh convergence theorem, prove separate results for convergence and for optimality, both under mild hypotheses.