Does $A^*$ being self-adjoint imply that $A$ is self adjoint?

Question

The question is in the title. This arose when I read that for some linear densely defined symmetric operator A that is not self-adjoint, and B a self-adjoint extension of $A$, we had $A \subsetneq B \subsetneq A^*$. The first strict inclusion seems clear because $A$ is not self-adjoint, but the second is not obvious to me.