Summary: A \(k\)-rainbow path in a graph with colored edges is a path of length \(k\) where each edge has a different color. In this note, we settle the problem of obtaining a constructive \(k\)-coloring of the edges of \(K_n\) in which one may find, between any pair of vertices, a large number of internally disjoint \(k\)-rainbow paths. In fact, our construction obtains the largest possible number of paths. This problem was considered in a less general setting by G. Chartrand, G.L. Johns, K.A. McKeon, and {|it P/ Zhang} [“On the rainbow connectivity of cages,” Congr. Numerantium 184, 209–222 (2007; Zbl 1138.05042)].