  Let $X, Y$ be two complex manifolds of dimension 1 which are countable at
infinity, let $D\subset X,$ $ G\subset Y$ be two open sets, let $A$ (resp. $B$)
be a subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold
cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in addition that
$D$ (resp. $G$) is {\it Jordan-curve-like on $A$} (resp. $B$) and that $A$ and
$B$ are {\it of positive length}. We determine the "envelope of holomorphy"
$\hat{W}$ of $W$ in the sense that any function locally bounded on $W,$
measurable on $A\times B,$ and separately holomorphic on $(A\times G) \cup
(D\times B)$ "extends" to a function holomorphic on the interior of $\hat{W}.$
