  We construct a complete convergent normal form for a real hypersurface in
$\CC{N},\,N\geq 2$ at generic Levi degeneracy. This seems to be the first
convergent normal form for a Levi-degenerate hypersurface. In particular, we
obtain, in the spirit of the work of Chern and Moser \cite{chern},
distinguished curves in the Levi degeneracy set, that we call \it degenerate
chains.
