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}Xe)=(1.4±6.6_{stat}±2.0_{syst})×10^{-28} e cm. This corresponds to an upper limit of |d_{A}(^{129}Xe)|<1.4×10^{-27} e cm (95% C.L.), a factor of 5 more sensitive than the limit set in 2001.