  We consider the space of countable structures with fixed underlying set in a
given countable language. We show that the number of ergodic probability
measures on this space that are $S_\infty$-invariant and concentrated on a
single isomorphism class must be zero, or one, or continuum. Further, such an
isomorphism class admits a unique $S_\infty$-invariant probability measure
precisely when the structure is highly homogeneous; by a result of Peter J.
Cameron, these are the structures that are interdefinable with one of the five
reducts of the rational linear order $(\mathbb{Q}, <)$.
