  We determine the number of labelled chordal planar graphs with $n$ vertices,
which is asymptotically $c_1\cdot n^{-5/2} \gamma^n n!$ for a constant $c_1>0$
and $\gamma \approx 11.89235$. We also determine the number of rooted simple
chordal planar maps with $n$ edges, which is asymptotically $c_2 n^{-3/2}
\delta^n$, where $\delta = 1/\sigma \approx 6.40375$, and $\sigma$ is an
algebraic number of degree 12. The proofs are based on combinatorial
decompositions and singularity analysis. Chordal planar graphs (or maps) are a
natural example of a subcritical class of graphs in which the class of
3-connected graphs is relatively rich. The 3-connected members are precisely
chordal triangulations, those obtained starting from $K_4$ by repeatedly adding
vertices adjacent to an existing triangular face.
