  We analyze the spectrum of the Neumann-Poincar\'e (NP) operator for a doubly
connected domain lying between two level curves defined by a conformal mapping,
where the inner boundary of the domain is of general shape. The analysis relies
on an infinite-matrix representation of the NP operator involving the Grunsky
coefficients of the conformal mapping and an application of the Gershgorin
circle theorem. As the thickness of the domain shrinks to zero, the spectrum of
the doubly connected domain approaches the interval $[-1/2,1/2]$ in the
Hausdorff distance and the density of eigenvalues approaches that of a thin
circular annulus.
