We establish a theorem computing the cohomology groups of line bundles on
homogeneous ind-varieties $G/B$ for diagonal ind-groups $G$. The main
difficulty in proving this analog of the classical Bott-Borel-Weil theorem is
in defining an appropriate analog $W_B$ of the Weyl group so that the action of
$W_B$ on weights of $G$ is compatible with the analog of the Demazure "action"
of the Weyl group on the cohomology of line bundles.