  We investigate partially disordered antiferromagnetism in
CoCl$_2$-2SC(NH$_2$)$_2$, in which a-b plane hexagonal layers are staggered
along the c-axis, rather than stacked. A robust 1/3 state forms in applied
magnetic fields which the spins are locked, varying neither as a function of
temperature nor field. By contrast, in zero field, partial antiferromagnetic
order occurs, in which free spins are available to create a Curie-like magnetic
susceptibility. We report measurements of the crystallographic structure, and
the specific heat, magnetization, and electric polarization down to $T$ = 50 mK
and up to $\mu_{0}H$ = 60 T. The Co$^{2+}$ $S = 3/2$ spins are Ising-like and
form distorted hexagonal layers. The Ising energy scale is well separated from
the magnetic exchange, and both energy scales are accessible to the
measurements allowing us to cleanly parameterize them. In transverse fields, a
quantum Ising phase transition can be observed at 2 T. Finally we find that
magnetic exchange striction induces changes in the electric polarization up to
3 $\mu$C/m$^2$ and single-ion magnetic anisotropy effects induce a much a
larger electric polarization change of 300 $\mu$C/m$^2$.
