  We consider the minimum-weight path between any pair of nodes of the n-vertex
complete graph in which the weights of the edges are i.i.d. exponentially
distributed random variables. We show that the longest of these minimum-weight
paths has about \alpha^* \log n$ edges where \alpha^* ~ 3.5911 is the unique
solution of the equation $alpha log(alpha) - \alpha =1. This answers a question
posed by Janson (1999).
