  Let $Z$ be a closed subscheme of a smooth complex projective variety
$Y\subseteq \Ps^N$, with $\dim\,Y=2r+1\geq 3$. We describe the intermediate
N\'eron-Severi group (i.e. the image of the cycle map $A_r(X)\to
H_{2r}(X;\mathbb{Z})$) of a general smooth hypersurface $X\subset Y$ of
sufficiently large degree containing $Z$.
