  The infinite-bin model is a one-dimensional particle system on $\mathbb{Z}$
introduced by Foss and Konstantopoulos in relation with last passage
percolation on complete directed acyclic graphs. In this model, at each integer
time, a particle is selected at random according to its rank, and produces a
child at the location immediately to its right. In this article, we consider
the limiting distribution of particles after an infinite number of branching
events have occurred. Under mild assumptions, we prove that the event (called
freezing) that a location contains only a finite number of balls satisfies a
$0-1$ law and we provide various criteria to determine whether freezing occurs.
