  The magnetic molecule Fe_8 has been observed to have a rich pattern of
degeneracies in its magnetic spectrum as the static magnetic field applied to
the molecule is varied. The points of degeneracy, or diabolical points in the
magnetic field space, are found exactly in the simplest model Hamiltonian for
this molecule. The points are shown to form a perfect centered rectangular
lattice, and are shown to be multiply diabolical in general. The multiplicity
is found. An earlier semiclassical solution to this problem is thereby shown to
be exact in leading order in 1/J where J is the spin.
