I examine the orbital evolution of eccentric binary systems consisting of a close Be star-neutron star pair. This is probably dominated by the interaction of the neutron star with the Be star and its wind near periastron. This can significantly change the orbital elements over the main-sequence lifetime of the Be star. The periastron distance remains constant during the evolution, as does the combination P(circ) = P(1 - e2)3/2 (where e is the eccentricity and P the orbital period), which is approximately equal to the period of the progenitor binary just before the supernova producing the neutron star. During orbital evolution the period and eccentricity increase or decrease together according as the neutron star gains (DELTAJ > 0) or loses (DELTAJ < 0) angular momentum at periastron. The very high orbital eccentricity e congruent-to 0.97 of the recently discovered millisecond pulsar-Be star binary PSR 1259-63 suggests that DELTAJ > 0 is possible in at least some cases. This can occur if, for example, the rotational velocity of the Be star's equatorial wind is super-Keplerian, as suggested by some models. PSR 1259-63 could then have descended from a Be-X-ray binary like A0538-66. Accretion would have spun up the neutron star to its present 47 ms spin period, making it detectable as a radio pulsar for much of its very long (P greater than or similar to 3-6 yr) orbit. This picture predicts that accretion occurs near periastron, extinguishing the radio emission, and that the system may become an X-ray source there. Determinations of the pulsar magnetic field should give a value characteristic of recycled pulsars (less than or similar to 10(10) G). Longer-period systems may exist, detectable only through the positional coincidence of an apparently isolated millisecond pulsar with a Be star.