We present new Gemini Planet Imager observations of the young exoplanet 51 Eridani b that provide further evidence that the companion is physically associated with 51 Eridani. Combining this new astrometric measurement with those reported in the literature, we significantly reduce the posterior probability that 51 Eridani b is an unbound foreground or background T-dwarf in a chance alignment with 51 Eridani to 2 × 10-7, an order of magnitude lower than previously reported. If 51 Eridani b is indeed a bound object, then we have detected orbital motion of the planet between the discovery epoch and the latest epoch. By implementing a computationally efficient Monte Carlo technique, preliminary constraints are placed on the orbital parameters of the system. The current set of astrometric measurements suggest an orbital semimajor axis of AU, corresponding to a period of years (assuming a mass of 1.75 Mo for the central star), and an inclination of deg. The remaining orbital elements are only marginally constrained by the current measurements. These preliminary values suggest an orbit that does not share the same inclination as the orbit of the distant M-dwarf binary, GJ 3305, which is a wide physically bound companion to 51 Eridani.