  The Wiener index of a connected graph is the sum of the distances between all
pairs of vertices in the graph. It was conjectured that the Wiener index of an
$n$-vertex maximal planar graph is at most
$\lfloor\frac{1}{18}(n^3+3n^2)\rfloor$. We prove this conjecture and for every
$n$, $n \geq 10$, determine the unique $n$-vertex maximal planar graph for
which this maximum is attained.
