  Let $X$ be a complex submanifold of dimension $d$ of $\mathbb
P^m\times\mathbb P^n$ ($m\geq n\geq 2$) and denote by $\alpha\colon\Pic(\mathbb
P^m\times\mathbb P^n)\to \Pic(X)$ the restriction map of Picard groups, by
$N_{X|\mathbb P^m\times\mathbb P^n}$ the normal bundle of $X$ in $\mathbb
P^m\times\mathbb P^n$. Set $t:=\max\{\dim\pi_1(X),\dim\pi_2(X)\}$, where
$\pi_1$ and $\pi_2$ are the two projections of $\mathbb P^m\times\mathbb P^n$.
We prove a Barth-Lefschetz type result as follows: {\em Theorem.} {\it If
$d\geq \frac{m+n+t+1}{2}$ then $X$ is algebraically simply connected, the map
$\alpha$ is injective and $\Coker(\alpha)$ is torsion-free. Moreover $\alpha$
is an isomorphism if $d\geq\frac{m+n+t+2}{2}$, or if $d=\frac{m+n+t+1}{2}$ and
$N_{X|\mathbb P^m\times\mathbb P^n}$ is decomposable.} These bounds are
optimal. The main technical ingredients in the proof are: the Kodaira-Le Potier
vanishing theorem in the generalized form of Sommese (\cite{LP}, \cite{ShS}),
the join construction and an algebraisation result of Faltings concerning small
codimensional subvarieties in $\mathbb P^N$ (see \cite{Fa}).
