  Let $R_n$ be the range of a critical branching random walk with $n$ particles
on $\mathbb Z^d$, which is the set of sites visited by a random walk indexed by
a critical Galton--Watson tree conditioned on having exactly $n$ vertices. For
$d\in\{3, 4, 5\}$, we prove that $n^{-\frac{d-2}4} \mathtt{cap}^{(d)}(R_n)$,
the renormalized capacity of $R_n$, converges in law to the capacity of the
support of the integrated super-Brownian excursion. The proof relies on a study
of the intersection probabilities between the critical branching random walk
and an independent simple random walk on $\mathbb Z^d$.
