  We study spin transport in a Hubbard chain with strong, random, on--site
potential and with spin--dependent hopping integrals, $t_{\sigma}$. For the the
SU(2) symmetric case, $t_{\uparrow} =t_{\downarrow}$, such model exhibits only
partial many-body localization with localized charge and (delocalized)
subdiffusive spin excitations. Here, we demonstrate that breaking the SU(2)
symmetry by even weak spin--asymmetry, $t_{\uparrow} \ne t_{\downarrow}$,
localizes spins and restores full many-body localization. To this end we derive
an effective spin model, where the spin subdiffusion is shown to be destroyed
by arbitrarily weak $t_{\uparrow} \ne t_{\downarrow}$. Instability of the spin
subdiffusion originates from an interplay between random effective fields and
singularly distributed random exchange interactions.
