  We study the relaxation of the exciton spin (longitudinal relaxation time
$T_{1}$) in single asymmetrical quantum dots due to an interplay of the
short--range exchange interaction and acoustic phonon deformation. The
calculated relaxation rates are found to depend strongly on the dot size,
magnetic field and temperature. For typical quantum dots and temperatures below
100 K, the zero--magnetic field relaxation times are long compared to the
exciton lifetime, yet they are strongly reduced in high magnetic fields. We
discuss explicitly quantum dots based on (In,Ga)As and (Cd,Zn)Se semiconductor
compounds.
