  We prove that the finite $\mathcal{W}$-algebra associated to
$\mathfrak{osp}_{1|2n}$ and its principal nilpotent element is isomorphic to
Gorelik's ghost center of $\mathfrak{osp}_{1|2n}$, which proves an analog of
Kostant's theorem for $\mathfrak{osp}_{1|2n}$.
