  We find using LSDA+U band structure calculations that the novel
one-dimensional cobaltate Ca$_3$Co$_2$O$_6$ is not a ferromagnetic half-metal
but a Mott insulator. Both the octahedral and the trigonal Co ions are formally
trivalent, with the octahedral being in the low-spin and the trigonal in the
high-spin state. The inclusion of the spin-orbit coupling leads to the
occupation of the minority-spin $d_{2}$ orbital for the unusually coordinated
trigonal Co, producing a giant orbital moment (1.57 $\mu_{B}$). It also results
in an anomalously large magnetocrystalline anisotropy (of order 70 meV),
elucidating why the magnetism is highly Ising-like. The role of the oxygen
holes, carrying an induced magnetic moment of 0.13 $\mu_{B}$ per oxygen, for
the exchange interactions is discussed.
