The helicity evolution of a neutrino interacting with randomly moving and polarized matter is studied. We derive the equation for the averaged neutrino helicity. The type of the neutrino interaction with background fermions is not fixed. In the particular case of a tau-neutrino interacting with ultrarelativistic electron-positron plasma we obtain the expression for the neutrino helicity relaxation rate in the explicit form. We study the neutrino spin relaxation in the relativistic primordial

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... sma. Supposing that the conversion of left-handed neutrinos into right-handed ones is suppressed at the early stages of the Universe evolution we get the upper limit on the tau-neutrino mass.