  We prove that each non-separable completely metrizable convex subset of a
Frechet space is homeomorphic to a Hilbert space. This resolves an old (more
than 30 years) problem of infinite-dimensional topology. Combined with the
topological classification of separable convex sets due to Klee, Dobrowoslki
and Torunczyk, this result implies that each closed convex subset of a Frechet
space is homemorphic to $[0,1]^n\times [0,1)^m\times l_2(k)$ for some cardinals
$0\le n\le\omega$, $0\le m\le 1$ and $k\ge 0$.
