  As introduced by Bollob\'as, a graph $G$ is weakly $H$-saturated if the
complete graph $K_n$ is obtained by iteratively completing copies of $H$ minus
an edge. For all graphs $H$, we obtain an asymptotic lower bound for the
critical threshold $p_c$, at which point the Erd\H{o}s--R\'enyi graph
${\mathcal G}_{n,p}$ is likely to be weakly $H$-saturated. We also prove an
upper bound for $p_c$, for all $H$ which are, in a sense, strictly balanced. In
particular, we improve the upper bound by Balogh, Bollob{\'a}s and Morris for
$H=K_r$, and we conjecture that this is sharp up to constants.
