  We prove that the random simple cubic planar graph $\mathsf{C}_n$ with an
even number $n$ of vertices admits a novel uniform infinite cubic planar graph
(UICPG) as quenched local limit. We describe how the limit may be constructed
by a series of random blow-up operations applied to the dual map of the
type~III Uniform Infinite Planar Triangulation established by Angel and Schramm
(Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation
between $\mathsf{C}_n$ and a model where the networks inserted at the links of
the largest $3$-connected component of $\mathsf{C}_n$ are replaced by
independent copies of a specific Boltzmann network. We prove that the number of
vertices of the largest $3$-connected component concentrates at $\kappa n$ for
$\kappa \approx 0.85085$, with Airy-type fluctuations of order $n^{2/3}$. The
second-largest component is shown to have significantly smaller size
$O_p(n^{2/3})$.
