  We study the spin-spin coupling between two single-electron quantum dots due
to the Coulomb and spin-orbit interactions, in the absence of tunneling between
the dots. We find an anisotropic XY spin-spin interaction that is proportional
to the Zeeman splitting produced by the external magnetic field. This
interaction is studied both in the limit of weak and strong Coulomb repulsion
with respect to the level spacing of the dot. The interaction is found to be a
non-monotonic function of inter-dot distance $a_0$ and external magnetic field,
and, moreover, vanishes for some special values of $a_0$ and/or magnetic field
orientation. This mechanism thus provides a new way to generate and tune spin
interaction between quantum dots. We propose a scheme to measure this spin-spin
interaction based on the spin-relaxation-measurement technique.
