  We measure the spin-resolved transport of dipolar excitons in a biased GaAs
double quantum well structure. From these measurements we extract both spin
lifetime and mobility of the excitons. We find that below a temperature of
$4.8$K, there is a sharp increase in the spin lifetime of the excitons,
together with a sharp reduction in their mobility. Below a critical power the
spin lifetime increases with increasing mobility and density, while above the
critical power the opposite trend is observed. We interpret this transition as
an evidence of the interplay between two different spin dephasing mechanisms:
at low mobility the dephasing is dominated by the hyperfine interaction with
the lattice nuclei spins, while at higher mobility the spin-orbit interaction
dominates, and a Dyakonov-Perel spin relaxation takes over. The excitation
power and temperature regime where the hyperfine interaction induced spin
dephasing is observed correlates with the regime where a dark dipolar quantum
liquid was reported recently on a similar sample.
