We present a study of the smallest and faintest galaxies found in a very deep photographic R band survey of regions of the Virgo Cluster, totalling over 3 square degrees, made with the UK Schmidt Telescope. The objects we detect have the same physical sizes and surface brightnesses as Local Group dwarf spheroidal galaxies. The luminosity function of these extremely low luminosity galaxies (down to M_R =~ -11 or about 5 X 10^{-5} L*) is very steep, with a power law slope alpha =~ -2, as would be expected in many theories of galaxy formation via hierarchical clustering, supporting previous observational evidence at somewhat higher luminosities in other clusters.

8 pages, LateX (uses AASTeX aas2pp4 style file, included), with one embedded postscript figure, also available at http://WWW.star.bris.ac.uk/publs_preprints/preprints.html, accepted for publication in the Astrophysical Journal Letters