  For 3-dimensional hyperbolic cone structures with cone angles $\theta$, local
rigidity is known for $0 \leq \theta \leq 2\pi$, but global rigidity is known
only for $0 \leq \theta \leq \pi$. The proof of the global rigidity by Kojima
is based on the fact that hyperbolic cone structures with cone angles at most
$\pi$ do not degenerate in deformations decreasing cone angles to zero.
  In this paper, we give an example of a degeneration of hyperbolic cone
structures with decreasing cone angles less than $2\pi$. These cone structures
are constructed on a certain alternating link in the thickened torus by gluing
four copies of a certain polyhedron. For this construction, we explicitly
describe the isometry types on such a hyperbolic polyhedron.
