  We study the random simple connected cubic planar graph $\mathsf{C}_n$ with
an even number $n$ of vertices. We show that the Brownian map arises as
Gromov--Hausdorff--Prokhorov scaling limit of $\mathsf{C}_n$ as $n \in 2 \ndN$
tends to infinity, after rescaling distances by $\gamma n^{-1/4} $ for a
specific constant $\gamma>0$.
