  We show that the Conway--Maxwell--Poisson distribution can be arbitrarily
underdispersed when parametrized via its mean. More precisely, if the mean
$\mu$ is an integer then the limiting distribution is a unit probability mass
at $\mu$. If the mean $\mu$ is not an integer then the limiting distribution is
a shifted Bernoulli on the two values $\floor{\mu}$ and $\ceil{\mu}$ with
probabilities equal to the fractional parts of $\mu$. In either case, the
limiting distribution is the most underdispersed discrete distribution possible
for any given mean. This is currently the only known generalization of the
Poisson distribution exhibiting this property. Four practical implications are
discussed, each adding to the claim that the (mean-parametrized)
Conway--Maxwell--Poisson distribution should be considered the default model
for underdispersed counts. We suggest that all future generalizations of the
Poisson distribution be tested against this property.
