We prove that the sweeping components of the space of smooth rational curves
in a smooth hypersurface of degree $d$ in $P^n$ are not uniruled if $(n+1)/2
\leq d \leq n-3$. We also show that for any positive integer $e$, the space of
smooth rational curves of degree $e$ in a general hypersurface of degree $d$ in
$P^n$ is not uniruled when $d \geq e \sqrt{n}$.