  We say that a weighted shift $W_\alpha$ with (positive) weight sequence
$\alpha: \alpha_0, \alpha_1, \ldots$ is {\it moment infinitely divisible} (MID)
if, for every $t > 0$, the shift with weight sequence $\alpha^t: \alpha_0^t,
\alpha_1^t, \ldots$ is subnormal. \ Assume that $W_{\alpha}$ is a contraction,
i.e., $0 < \alpha_i \le 1$ for all $i \ge 0$. \ We show that such a shift
$W_\alpha$ is MID if and only if the sequence $\alpha$ 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.
