We report on the discovery of a faint (MV~-10.6+/-0.2) dwarf spheroidal galaxy on deep F606W and F814W Hubble Space Telescope images of a Virgo intracluster field. The galaxy is easily resolved in our images, as our color magnitude diagram (CMD) extends >~1 magnitude beyond the tip of the red giant branch (RGB). Thus, it is the deepest CMD for a small dwarf galaxy inside a cluster environment. Using the colors of the RGB stars, we derive a metal abundance for the dwarf of [M/H]=-2.3+/-0.3 and show that the metallicity dispersion is less than 0.6 dex at 95% confidence. We also use the galaxy's lack of AGB stars and the absence of objects brighter than Mbol~-4.1+/-0.2 to show that the system is old (t>~10 Gyr). Finally, we derive the object's structural parameters and show that the galaxy displays no obvious evidence of tidal threshing. Since the tip of the red giant branch distance [(m-M)0=31.23+/-0.17 or D=17.6+/-1.4 Mpc] puts the galaxy near the core of the Virgo cluster, one might expect the object to have undergone some tidal processing. Yet the chemical and morphological similarity between the dwarf and the dSph galaxies of the Local and M81 Group demonstrates that the object is indeed pristine and not the shredded remains of a much larger galaxy. We discuss the possible origins of this galaxy and suggest that it is just now falling into Virgo for the first time.