  Let $p_n^G$ be the probability for a planar Poisson-Voronoi cell to be
$n$-sided {\it and\,} have only Gabriel neighbors. Using an exact coordinate
transformation followed by scaling arguments and a mean-field type calculation,
we obtain the asymptotic expansion of $\log p_n^G$ in the limit of large $n$.
We determine several statistical properties of a many-sided cell obeying this
`Gabriel condition.' In particular, the cell perimeter, when parametrized as a
function $\tau(\theta)$ of the polar angle $\theta$, behaves as a Brownian
bridge on the interval $0\le\theta\le 2\pi$. We point out similarities and
differences with related problems in random geometry.
