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.