  We study the asymptotic behaviour of the eigenvalues of the Laplace-Beltrami
operator on a compact hypersurface in \mathds{R}^{n+1} as it is flattened into
a singular double-sided flat hypersurface. We show that the limit spectral
problem corresponds to the Dirichlet and Neumann problems on one side of this
flat (Euclidean) limit, and derive an explicit three-term asymptotic expansion
for the eigenvalues where the remaining two terms are of orders
\varepsilon^2\log\varepsilon and \varepsilon^2.
