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

Magnetic phase transitions and spin density distribution in the molecular multiferroic GaV$_4$S$_8$ system