We present GALEX near-ultraviolet (2310 Å) and far-ultraviolet (1530 A) images of the interacting galaxy NGC 4438 (Arp 120) in the center of the Virgo Cluster. These images show an extended (20 kpc) tidal tail at the northwest edge of the galaxy that was previously undetected at other wavelengths; this tail is 15-25 kpc from NGC 4438's nucleus. Except for in the nucleus, the UV morphology of NGC 4438 is totally different from the Ha + [N II] morphology, which is more similar to the X-ray emission, confirming its gas cooling origin. We study the star formation history of NGC 4438 by combining spectrophotometric data in the UV-visible-near-IR wavelength range with population synthesis and galaxy evolution models. The data are consistent with a recent (~10 Myr), instantaneous burst of star formation in the newly discovered UV northwestern tail that is significantly younger man the age of the tidal interaction with NGC 4435, dated by dynamical models at ~100 Myr ago. Recent star formation events are also present at the edge of the northern arm and in the southern tail, while totally lacking in the other regions, which are dominated by the old stellar population that was perturbed during the dynamical interaction with NGC 4435. The contribution of this recent starburst to the total galaxy stellar mass is lower than 0.1%, an extremely low value for such a violent interaction. High-velocity, off-center tidal encounters such as that observed in Arp 120 are thus not sufficient to significantly increase the star formation activity of cluster galaxies. © 2005. The American Astronomical Society. All rights reserved.