Rescaled bipartite planar maps converge to the Brownian map

For every integer $n\ge 1$, we consider a random planar map ${\mathcal{M}}_n$ which is uniformly distributed over the class of all rooted bipartite planar maps with $n$ edges. We prove that the vertex set of ${\mathcal{M}}_n$ equipped with the graph distance rescaled by the factor $(2n{)}^{-1\mathrm{/}4}$ converges in distribution, in the Gromov-Hausdorff sense, to the Brownian map. This complements several recent results giving the convergence of various classes of random planar maps to the Brownian map.