[For part I see Adv. Math. 69, No.1, 115-132 (1988; Zbl 0646.60017).] Let G be a commutative topological group which is separated by its continuous characters. Consider a triangular array of distributions on G with the property that its row-wise convolution products converge to a limit \(\mu\). The main result states that, under an infinitesimality condition, \(\mu\) is weakly infinitely divisible in the sense that for each \(n\in {\mathbb{N}}\) there exist a distribution \(\mu_ n\) and an element \(g_ n\) of G such that \(\mu\) is the n-th convolution power of \(\mu_ n\), shifted by \(g_ n\).