  In this paper we report strong evidence for a ~304-day periodicity in the
spin history of the accretion-powered pulsar GX 1+4 that is very likely to be a
signature of the orbital period of the system. Using BATSE public-domain data,
we show a highly-significant periodic modulation of the pulsar frequency from
1991 to date which is in excellent agreement with the ephemeris proposed by
Cutler, Dennis & Dolan in 1986, which were based on a few events of enhanced
spin-up that occurred during the pulsar's spin-up era in the 1970s. Our results
indicate that the orbital period of GX 1+4 is 303.8+-1.1 days, making it by far
the widest low-mass X-ray binary system known. 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.
