  Let $M$ be a connected real-analytic hypersurface in $\C^N$ and $\S$ the unit
real sphere in $\C^{N'}$, $N'> N\geq 2$. Assume that $M$ does not contain any
complex-analytic hypersurface of $\C^N$ and that there exists at least one
strongly pseudoconvex point on $M$. We show that any CR map $f\colon M\to \S$
of class $C^{N'-N+1}$ extends holomorphically to a neighborhood of $M$ in
$\C^N$.
