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We report results of a new technique to measure the electric dipole moment of 129Xe with ³He 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 dA(129Xe) = (1.4 ± 6.6stat ± 2.0syst) × 10-28 e cm. This corresponds to an upper limit of |dA(129Xe)| < 1.4 × 10-27 e cm (95% C.L.), a factor of 5 more sensitive than the limit set in 2001. [ABSTRACT FROM AUTHOR]