  We calculate the acoustic phonon-assisted exciton spin relaxation in single
self-assembled In$_{1-x}$Ga$_x$As/GaAs quantum dots using an atomic empirical
pseudopotential method. We show that the transition from bright to dark exciton
states is induced by Coulomb correlation effects. The exciton spin relaxation
time obtained from sophisticated configuration interaction calculations is
approximately 15--55 $\mu$s in pure InAs/GaAs QDs and even longer in alloy
dots. These results contradict previous theoretical and experimental results,
which suggest very short exciton spin times (a few ns), but agree with more
recent experiments that suggest that excitons have long spin relaxation times
($>$ 1 $\mu$s).
