  We identify the asymptotic probability of a configuration model
$\mathrm{CM}_n(\boldsymbol{d})$ to produce a connected graph within its
critical window for connectivity that is identified by the number of vertices
of degree 1 and 2, as well as the expected degree. In this window, the
probability that the graph is connected converges to a non-trivial value, and
the size of the complement of the giant component weakly converges to a finite
random variable. Under a finite second moment condition we also derive the
asymptotics of the connectivity probability conditioned on simplicity, from
which the asymptotic number of simple connected graphs with a prescribed degree
sequence follows.
