A solvable Hamburger moment problem has a unique solution or infinitely many

Let $m=(m_n)_{n\ge0}$ be a moment sequence, i.e. its Hamburger moment problem has at least one solution. Then at least one of the following holds:

1. the problem has a unique solution $\mu$;
2. the set of solutions $\{\mu : \mu \text{ positive Borel measure on }\mathbb R,\ \int x^n\,d\mu=m_n\ \forall n\}$ is infinite.

Formalization Note The statement is the disjunction "unique solution or infinitely many solutions"; the two alternatives are automatically exclusive.