  Let $G$ be a graph in which each vertex initially has weight 1. In each step,
the unit weight from a vertex $u$ to a neighbouring vertex $v$ can be moved,
provided that the weight on $v$ is at least as large as the weight on $u$. The
unit acquisition number of $G$, denoted by $a_u(G)$, is the minimum cardinality
of the set of vertices with positive weight at the end of the process (over all
acquisition protocols). In this paper, we investigate the Erd\H{o}s-R\'{e}nyi
random graph process $(\mathcal{G}(n,m))_{m =0}^{N}$, where $N = {n \choose
2}$. We show that asymptotically almost surely $a_u(\mathcal{G}(n,m)) = 1$
right at the time step the random graph process creates a connected graph.
Since trivially $a_u(\mathcal{G}(n,m)) \ge 2$ if the graphs is disconnected,
the result holds in the strongest possible sense.
