  A limit on the mass of the tau neutrino is derived from 4.5 million tau pairs
produced in an integrated luminosity of 5.0 fb^{-1} of electron-positron
annihilation to tau pairs at center of mass energies near 10.6 GeV. The
measurement technique involves a two-dimensional extended likelihood analysis,
including the dependence of the end-point population on the 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 as well as the decay to
three charged pions, two neutral pions and a neutrino to obtain an upper limit
of 30 MeV/c^2 at 95% C.L.
