We have used the WFPC2 camera of the Hubble Space Telescope to obtain deep F814W images of a blank field in the Virgo Cluster located 41 arcmin northwest of M87. We perform star counts in that field, and in another Virgo field observed by Ferguson, Tanvir & von Hippel (1998), and show that, when compared to the Hubble Deep Field North and South, the Virgo Cluster contains an excess of objects with magnitudes I > 27. We attribute this excess to a population of intracluster red-giant branch (IC-RGB) stars. By modeling the luminosity function of these stars, we show that the tip of the Virgo RGB is at I = 27.31 +0.27/-0.17 and that the cluster contains a small, but significant, excess of stars that are up to ~1 mag brighter than this tip. If this luminous component is due entirely to stars on the asymptotic giant branch (AGB), it implies an age for the population of > 2 Gyr; if foreground RGB stars contribute to the luminous tail, then the derived age for the stars is older still. The luminosity function also suggests that most of the intracluster stars are moderately metal-rich (-0.8 < [Fe/H] <-0.2), a result consistent with that expected from stars that have been tidally stripped from intermediate luminosity galaxies. Additionally, a comparison with the planetary nebulae in our field also supports this view, although the existence of a more metal-poor population (from stripped dwarfs) cannot be ruled out. Our derived average surface brightness, mu_I = 27.9 +0.3/-0.5 mag/arcsec^2 for Virgos diffuse component suggests that intracluster stars contribute 10% to 20% of the clusters total I-band luminosity.
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