  Estimates are made of the ultra-high energy neutrino cross sections based on
an extrapolation to very small Bjorken x of the logarithmic Froissart
dependence in x shown previously to provide an excellent fit to the measured
proton structure function F_2^p(x,Q^2) over a broad range of the virtuality
Q^2. Expressions are obtained for both the neutral current and the charged
current cross sections. Comparison with an extrapolation based on perturbative
QCD shows good agreement for energies where both fit data, but our rates are as
much as a factor of 10 smaller for neutrino energies above 10^9 GeV, with
important implications for experiments searching for extra-galactic neutrinos.
