We consider the Cauchy problem
\begin{alignat*}{2}
&u_t=u_{xx}+f(u), &\qquad &x\in \mathbb R,\ t>0,\\
& u(x,0) =u_0(x),& &x\in \mathbb R,
\end{alignat*}
where $f$ is a $C^1$ function on $\mathbb R$ with $f(0)=0$, and
$u_0$ is a nonnegative continuous function on $\mathbb R$ whose
limits at $\pm\infty$ are equal to 0. Assuming that the
solution $u$ is bounded, we study its asymptotic behavior as
$t\to\infty$. In the first part of this study,
we proved a general quasiconvergence result: as $t\to\infty$,
the solution approaches a set of steady states in the topology of
$L^\infty_{loc}(\mathbb R)$. In this paper, we show that under certain
generic, explicitly formulated conditions on the nonlinearity $f$,
the solution necessarily converges to a single steady steady $\varphi$
in $L^\infty_{loc}(\mathbb R)$. Then, under the same
conditions, we describe the global asymptotic shape of the
solution: the graph of $u(\cdot,t)$ has a top part
close to the graph of $\varphi$ and two sides
taking shapes of ``terraces'' moving
in the opposite directions with precisely determined speeds.