We consider the problem of partitioning the nodes of a complete edge weighted graph into k clusters so as to minimize the sum of the diameters of the clusters. Since the problem is NP-complete, our focus is on the development of good approximation algorithms. When edge weights satisfy the triangle inequality, we present the first approximation algorithm for the problem. The approximation algorithm yields a solution that has no more than 10k clusters such that the total diameter of these clusters is within a factor O(log (n/k)) of the optimal value for k clusters, where n is the number of nodes in the complete graph. For any fixed k, we present an approximation algorithm that produces k clusters whose total diameter is at most twice the optimal value. When the distances are not required to satisfy the triangle inequality, we show that, unless P = NP, for any n > 1, there is no polynomial time approximation algorithm that can provide a performance guarantee of n even when the number of clusters is fixed at 3. Other results obtained include a polynomial time algorithm for the problem when the underlying graph is a tree with edge weights.