  We use the results of Maher {\em et al.\/} (preceding paper) to construct the
matrix of $j=0$ partial-wave two-body and $2\rightarrow3$ scattering amplitudes
for the scattering of longitudinally polarized gauge bosons $W_L^\pm$, $Z_L$
and Higgs bosons $H$ correct to two loops in the high-energy, heavy-Higgs limit
$\sqrt{s}\gg M_H\gg M_W$. We show explicitly that the energy dependence of the
$2\rightarrow2$ amplitudes can be completely absorbed into a running quartic
Higgs coupling $\lambda_s= \lambda_s(s,M_H^2)$ and factors which involve small
anomalous dimensions and remain near unity. After diagonalizing the matrix of
partial-wave amplitudes, we use an Argand-diagram analysis to show that the
elastic scattering amplitudes are approximately unitary and weakly interacting
for $\lambda_s\alt2.3$, but that three-loop corrections are necessary to
restore unitarity for larger values of $\lambda_s$. That is, the interactions
in the Higgs sector of the standard model are effectively strong with respect
to the perturbative expansion for $\lambda_s\agt2.3$. The bound
$\lambda_s\alt2.3$ for a weakly interacting theory translates to a physical
Higgs mass $M_H\alt380$ GeV if the bound is to hold for energies up to a few
TeV, or $M_H\leq155$ GeV in perturbatively unified theories with mass scales of
order $10^{16}$ GeV.
