  Given a hypersurface $M$ of null scalar curvature in the unit sphere
$\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank
greater than 2, we construct a singular scalar-flat hypersurface in $\Rr^{n+1}$
as a normal graph over a truncated cone generated by $M$. Furthermore, this
graph is 1-stable if the cone is strictly 1-stable.
