  We achieve a detailed understanding of the $n$-sided planar Poisson-Voronoi
cell in the limit of large $n$. Let ${p}\_n$ be the probability for a cell to
have $n$ sides. We construct the asymptotic expansion of $\log {p}\_n$ up to
terms that vanish as $n\to\infty$. We obtain the statistics of the lengths of
the perimeter segments and of the angles between adjoining segments: to leading
order as $n\to\infty$, and after appropriate scaling, these become independent
random variables whose laws we determine; and to next order in $1/n$ they have
nontrivial long range correlations whose expressions we provide. The $n$-sided
cell tends towards a circle of radius $(n/4\pi\lambda)^{\half}$, where
$\lambda$ is the cell density; hence Lewis' law for the average area $A\_n$ of
the $n$-sided cell behaves as $A\_n \simeq cn/\lambda$ with $c=1/4$. For
$n\to\infty$ the cell perimeter, expressed as a function $R(\phi)$ of the polar
angle $\phi$, satisfies $d^2 R/d\phi^2 = F(\phi)$, where $F$ is known Gaussian
noise; we deduce from it the probability law for the perimeter's long
wavelength deviations from circularity. Many other quantities related to the
asymptotic cell shape become accessible to calculation.
