  We prove that every biregular automorphism of the affine algebraic variety
${\mathbb P}^M\setminus S$, $M\geqslant 3$, where $S\subset {\mathbb P}^M$ is a
hypersurface of degree $m\geqslant M+1$ with a unique singular point of
multiplicity $(m-1)$, resolved by one blow up, is a restriction of some
automorphism of the projective space ${\mathbb P}^M$, preserving the
hypersurface $S$; in particular, for a general hypersurface $S$ the group
$\mathop{\rm Aut} ({\mathbb P}^M\setminus S)$ is trivial.
