  Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect.
It is known that if delta(X) is at least 4, then X is isomorphic to a product
SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the
case where delta(X)=3. We show that there exists a smooth Fano variety T with
dim T=dim X-2 such that X is obtained from T with two possible explicit
constructions; in both cases there is a P^2-bundle Z over T such that X is the
blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2
subvarieties. Then we apply the structure theorem to Fano 4-folds, to the case
where X has Picard number 5, and to Fano varieties having an elementary
divisorial contraction sending a divisor to a curve. In particular we complete
the classification of Fano 4-folds with delta(X)=3.
