  The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a
graph $H$ is the maximum number of edges in an $n$-vertex planar graph without
$H$ as a subgraph. Let $C_{\ell}$ denote the cycle of length $\ell$. The planar
Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_{\ell})$ behaves differently for
$\ell\le 10$ and for $\ell\ge 11$, and it is known when $\ell \in \{3,4,5,6\}$.
We prove that $\textrm{ex}_{\mathcal P}(n,C_7) \le \frac{18n}{7} -
\frac{48}{7}$ for all $n > 38$, and show that equality holds for infinitely
many integers $n$.
