  We describe a new technique to measure the EDM of $^{129}$Xe with $^3$He
comagnetometry. Both species are polarized using spin-exchange optical pumping,
transferred to a measurement cell, and transported into a magnetically shielded
room, where SQUID magnetometers detect free precession in applied electric and
magnetic fields. The result of a one week run combined with a detailed study of
systematic effects is $d_A(^{129}\mathrm{Xe}) = (0.26 \pm 2.33_\mathrm{stat}
\pm 0.72_\mathrm{syst})\times10^{-27}~e\,\mathrm{cm}$. This corresponds to an
upper limit of $|d_A(^{129}\mathrm{Xe})| < 4.81\times 10^{-27}
~e\,\mathrm{cm}~(95\%~\mathrm{CL})$, a factor of 1.4 more sensitive than the
previous limit.
