  Let X be a Q-factorial Gorenstein Fano variety. Suppose that the
singularities of X are canonical and that the locus where they are non-terminal
has dimension zero. Let D be a prime divisor of X. We show that rho_X - rho_D <
9 (where rho is the Picard number). Moreover, if rho_X - rho_D > 3, there
exists a finite morphism from X to S x Y, where S is a surface with rho_S at
most 9. As an application we prove that, if X has dimension 3, then rho_X is at
most 10.
