In Part I (in this journal) we argued that the structure function $F_2^{gamma p}(x,Q^2)$ in deep inelastic $ep$ scattering, regarded as a cross section for virtual $gamma^*p$ scattering, has a saturated Froissart-bounded form behaving as $ln^2 (1/x)$ at small $x$. This form provides an excellent fit to the low $x$ HERA data, including the very low $Q^2$ regions, and can be extrapolated reliably to small $x$ using the natural variable $ln(1/x)$. We used our fit to derive quark distributions for values of $x$ down to $x=10^{-14}$. We use those distributions here to evaluate ultra-high energy (UHE) cross sections for neutrino scattering on an isoscalar nucleon, $N=(n+p)/2$, up to laboratory neutrino energies $E_ u sim 10^{16}$-$10^{17}$ GeV where there are now limits on neutrino fluxes. We estimate that these cross sections are accurate to $sim$2% at the highest energies considered, with the major uncertainty coming from the errors in the parameters that were needed to fit $F_2^{gamma p}(x,Q^2)$. We compare our results to recently published neutrino cross sections derived from NLO parton distribution functions, which become much larger at high energies because of the use of power-law extrapolations of quark distributions to small $x$. We argue that our calculation of the UHE $ u N$ cross sections is the best one can make based the existing experimental deep inelastic scattering data. Further, we show that the strong interaction Froissart bound of $ln^2 (1/x)$ on $F_2^{gamma p}$ translates to an exact bound of $ln^3E_ u$ for leading-order-weak $ u N$ scattering. The energy dependence of $ u N$ total cross section measurements consequently has important implications for hadronic interactions at enormous cms (center-of-mass) energies not otherwise accessible.
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