[For part I see Discrete Appl. Math. 6, 55-62 (1983; Zbl 0509.90030).] Let G be a graph with possibly multiple edges and let \((s_ 1,t_ 1)\), \((s_ 2,t_ 2)\), \((s_ 3,t_ 3)\) be pairs of vertices of G. We prove that if for each \(i=1,2,3\), there exist three edge-disjoint paths between \(s_ i\) and \(t_ i\), then there exist edge-disjoint paths \(P_ 1\), \(P_ 2\), \(P_ 3\) such that \(P_ i\) has ends \(s_ i\) and \(t_ i\) \((i=1,2,3)\), and prove that there exist flows between \(s_ i\) and \(t_ i\) of value 1 \((i=1,2,3)\) such that the total flow through each edge does not exceed 1, if and only if the obvious connectivity requirements are satisfied.