  We report strong evidence for a ~304-day periodicity in the spin history of
the accretion-powered pulsar GX1+4 that is most probably associated with the
orbital period of the system. We have used data from the Burst and Transient
Source Experiment on the Compton Gamma Ray Observatory to show a clear periodic
modulation of the pulsar frequency from 1991 to date, in excellent agreement
with the ephemeris proposed by Cutler, Dennis & Dolan (1986). Our results
indicate that the orbital period of GX1+4 is 303.8 +- 1.1 days, making it the
widest known low-mass X-ray binary system by more than one order of magnitude
and putting this long-standing question to rest. A likely scenario for this
system is an elliptical orbit in which the neutron star decreases its spin-down
rate (or even exhibits a momentary spin-up behavior) at periastron passages due
to the higher torque exerted by the accretion disk onto the magnetosphere of
the neutron star. These results are not inconsistent with both the X-ray pulsed
flux light curve measured by BATSE during the same epoch and the X-ray flux
history from the All-Sky Monitor (ASM) onboard the Rossi X-Ray Timing Explorer.
