  We show that for a smooth hypersurface $X\subset \bbP^n$ of degree at least
2, there exist arithmetically Cohen-Macaulay (ACM) codimension two subvarieties
$Y\subset X$ which are not an intersection $X\cap{S}$ for a codimension two
subvariety $S\subset\bbP^n$. We also show there exist $Y\subset X$ as above for
which the normal bundle sequence for the inclusion $Y\subset X\subset\bbP^n$
does not split.
