  Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset
D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N
A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times
A_N\subset\Bbb C^{n_1}\times...\times\Bbb C^{n_N}=\Bbb C^n. $$ Let
$U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a
relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $\Sigma_j$ be the
set of all $(z',z'')\in(A_1\times...\times A_{j-1})
\times(A_{j+1}\times...\times A_N)$ for which the fiber
$M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M\}$ is not
pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put $$
X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j
\times(A_{j+1}\times...\times A_N)\: (z',z'')\notin\Sigma_j\}. $$ Then there
exists a relatively closed pluripolar subset $\hat M\subset\hat X$ of the
`envelope of holomorphy' $\hat X\subset\Bbb C^n$ of $X$ such that: $\hat M\cap
X'\subset M$, for every function $f$ separately holomorphic on $X\setminus M$
there exists exactly one function $\hat f$ holomorphic on $\hat X\setminus\hat
M$ with $\hat f=f$ on $X'\setminus M$, and $\hat M$ is singular with respect to
the family of all functions $\hat f$. Some special cases were previously
studied in \cite{Jar-Pfl 2001c}.
