  We report the discovery of a ~500 kpc HI extension southwest of the Virgo
Cluster HI-rich pair NGC 4532/DDO 137, detected as part of the Arecibo Legacy
Fast ALFA (ALFALFA) Survey. The feature is the longest and most massive HI tail
structure so far found in the Virgo Cluster and, at 1.8 Mpc from M87, the most
distant from the main concentration of the intracluster medium. The structure
is spatially and spectrally separated into two ridges and is defined by diffuse
emission and discrete clumps of mass 2.5 - 6.8 x 10**7 solar masses. All
emission is blue-shifted with respect to the NGC 4532/DDO 137 pair emission.
Including diffuse emission, the structure has a total mass of up to 7 x 10**8
solar masses, equivalent to ~10% of the system's HI mass. Optical R-band
imaging finds no counterparts to a level of 26.5 mag arcsec**-2. The
characteristics of the structure appear most consistent with a tidal origin.
