We say that a weighted shift $$W_\alpha $$ W α with (positive) weight sequence $$\alpha : \alpha _0, \alpha _1, \ldots $$ α : α 0 , α 1 , … is moment infinitely divisible (MID) if, for every $$t > 0$$ t > 0 , the shift with weight sequence $$\alpha ^t: \alpha _0^t, \alpha _1^t, \ldots $$ α t : α 0 t , α 1 t , … is subnormal. Assume that $$W_{\alpha }$$ W α is a contraction, i.e., $$0 < \alpha _i \le 1$$ 0 < α i ≤ 1 for all $$i \ge 0$$ i ≥ 0 . We show that such a shift $$W_\alpha $$ W α is MID if and only if the sequence $$\alpha $$ α is log completely alternating. This enables the recapture or improvement of some previous results proved rather differently. We derive in particular new conditions sufficient for subnormality of a weighted shift, and each example contains implicitly an example or family of infinitely divisible Hankel matrices, many of which appear to be new.