  Motivated by the recent experiments that reported signatures of many-body
localization of ultracold atoms in optical lattices [M. Schreiber {\it et al.},
Science {\bf 349}, 842 (2015)], we study dynamics of highly excited states in
the strongly disordered Hubbard model in one dimension. Owing to the $SU(2)$
spin symmetry, spin degrees of freedom form a delocalized thermal bath with a
narrow bandwidth. The spin bath mediates slow particle transport, eventually
leading to delocalization of particles. The particle hopping rate is
exponentially small in $t/W$ ($t$, $W$ being hopping and disorder scales) owing
to the narrow bandwidth of the spin bath. We find the optimal lenghtscale for
particle hopping, and show that the particle transport rate depends strongly on
the density of singly occupied sites in the initial state. The delocalization
rate is zero for initial states with only doubly occupied or empty sites,
suggesting that such states are truly many-body localized, and therefore the
Hubbard model may host both localized and delocalized states. Full many-body
localization can be induced by breaking spin rotational symmetry.
