  We study the structure and melting of a classical bilayer system of dipoles,
in a setup where the dipoles are oriented perpendicular to the planes of the
layers and the density of dipoles is the same in each layer. Due to the
anisotropic character of the dipole-dipole interactions, we find that the
ground-state configuration is given by two hexagonal crystals positioned on top
of each other, independent of the interlayer spacing and dipolar density. For
large interlayer distances these crystals are independent, while in the
opposite limit of small interlayer distances the system behaves as a
two-dimensional crystal of paired dipoles. Within the harmonic approximation
for the phonon excitations, the melting temperature of these crystalline
configurations displays a non-monotonic dependence on the interlayer distance,
which is associated with a re-entrant melting behavior in the form of
solid-liquid-solid-liquid transitions at fixed temperature.

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