  We report results of a new technique to measure the electric dipole moment 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 from a one
week measurement campaign in 2017 and a 2.5 week campaign in 2018, combined
with detailed study of systematic effects, is $d_A(^{129}\mathrm{Xe}) = (1.4
\pm 6.6_\mathrm{stat} \pm 2.0_\mathrm{syst})\times10^{-28}~e\,\mathrm{cm}$.
This corresponds to an upper limit of $|d_A(^{129}\mathrm{Xe})| < 1.4 \times
10^{-27} ~e\,\mathrm{cm}~(95\%~\mathrm{CL})$, a factor of five more sensitive
than the limit set in 2001.
