  The collection of all $n$-point metric spaces of diameter $\le 1$ constitutes
a polytope $\mathcal{M}_n \subset \mathbb{R}^{\binom{n}{2}}$, called the
\emph{Metric Polytope}. In this paper, we consider the best approximations of
$\mathcal{M}_n$ by ellipsoids. We give an exact explicit description of the
largest volume ellipsoid contained in $\mathcal{M}_n$. When inflated by a
factor of $\Theta(n)$, this ellipsoid contains $\mathcal{M}_n$. It also turns
out that the least volume ellipsoid containing $\mathcal{M}_n$ is a ball. When
shrunk by a factor of $\Theta(n)$, the resulting ball is contained in
$\mathcal{M}_n$. We note that the general theorems on such ellipsoid posit only
that the pertinent inflation/shrinkage factors can be made as small as
$O(n^2)$.
