  We have carried out neutron diffraction and small angle neutron scattering
measurements on a high quality single crystal of the cubic lacunar spinel
multiferroic, GaV$_4$S$_8$, as a function of magnetic field and temperature to
determine the magnetic properties for the single electron that is located on
the tetrahedrally coordinated V$_4$ molecular unit. Our results are in good
agreement with the structural transition at 44 K from cubic to rhombohedral
symmetry where the system becomes a robust ferroelectric, while long range
magnetic order develops below 13 K in the form of an incommensurate cycloidal
magnetic structure, which can transform into a N\'eel-type skyrmion phase in a
modest applied magnetic field. Below 5.9(3) K, the crystal enters a
ferromagnetic phase, and we find the magnetic order parameter indicates a long
range ordered ground state with an ordered moment of 0.23(1) ${\mu}_\mathrm{B}$
per V ion. Both polarized and unpolarized neutron data in the
ferroelectric-paramagnetic phase have been measured to determine the magnetic
form factor. The data are consistent with a model of the single spin being
uniformly distributed across the V$_4$ molecular unit, rather than residing on
the single apical V ion, in substantial agreement with the results of
first-principles theory. In the magnetically ordered state, polarized neutron
measurements are important since both the cycloidal and ferromagnetic order
parameters are clearly coupled to the ferroelectricity, causing the structural
peaks to be temperature and field dependent. For the ferromagnetic ground
state, the spins are locked along the $[1,1,1]$ direction by a surprisingly
large anisotropy.
