  Let $\mathscr{X}\to W$ be a flat family of generically irreducible
hypersurfaces of degree $d\geq 2$ in $\PP^n$ with singular locus of dimension
$t$, with $W$ unirational of dimension $r$. We prove that if $n$ is large
enough with respect to $d$, $r$ and $t$, then $\mathscr{X}$ is unirational.
This extends results in \cite {Pre, HMP}.
