. Let Y be a Fano manifold of dimension n 3 with b 2 (Y ) = 1 and index n \Gamma 1 and let A be a projective manifold which is a double cover of Y . We determine which complex projective manifolds can admit A among their hyperplane sections. 0. Introduction and statement of the result The following is a natural problem in the study of hyperplane sections of projective varieties. Let Y be a complex projective manifold of dimension n and let : A ! Y be a double cover branched along a smooth hypersurface of Y . Assuming that Y is special enough, one has that A is also rather special. Then, according to the well known general philosophy that a projective manifold is as special as any of its hyperplane sections [S, p. 56], one can expect to be possible to classify all smooth projective (n + 1)-folds X ae P N admitting A among their hyperplane sections. This is the same as (*) classifying pairs (X; L) where X is a smooth projective (n + 1)-fold and L 2 Pic(X) is a very ample line bund..