  A limit on the mass of the tau neutrino is derived from 4.5x10^6 tau pairs
produced at a center of mass energy of 10.6 GeV. The measurement technique
involves a two-dimensional extended likelihood analysis, including the
dependence of the end-point population on the tau neutrino mass, and allows for
the first time an explicit background contribution. We use the decays of the
tau to five charged pions and a neutrino and to 3 charged pions, 2 neutral
pions and a neutrino to obtain an upper limit of 30 MeV, as well as a
preliminary limit of 31 MeV from the decay of a tau to 3 charged hadrons, a
neutral pi and a neutrino.
