Pulse frequencies from the archival ROSAT observations of 1E 2259+586 are presented. A short-term spin-up episode superposed on the secular spin-down trend is observed. Combining the previously published pulse frequencies, we estimate the variance of the angular velocity over a 15 yr span of observations. The level of fluctuations in angular velocity is similar to that of established accreting binary neutron stars. These results favor accretion by a neutron star from a very low mass companion as the source of the X-ray luminosity, although accretion by a solitary neutron star from a residual disk might be possible.