  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 $E_\nu \sim 10^{16}$-$10^{17}$
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 $\nu N$ 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 $\ln^3E_\nu$ for leading-order-weak $\nu N$
scattering. The energy dependence of $\nu N$ total cross section measurements
consequently has important implications for hadronic interactions at enormous
cms (center-of-mass) energies not otherwise accessible.
