  In this paper we will show that the generalized connected sum construction
for constant scalar curvature metrics can be extended to the zero scalar
curvature case. In particular we want to construct solutions to the Yamabe
equation on the generalized connected sum M = M_1 (\sharp_K) M_2 of two compact
Riemannian manifolds (M_1,g_1) and (M_2,g_2) with zero constant scalar
curvature along a common (isometrically embedded) submanifold (K,g_K) of
codimension \geq 3. We present here two kinds of construction. The first one is
the basic model and it works for every couple of scalar flat manifolds, but it
has a drawback. In fact following this method we are not allowed to choose a
scalar flat metric on the generalized connected sum, although the error can be
chosen as small as we want. The second construction is an adjustment of the
first one which enable us to get a zero scalar curvature metric on the final
manifold, but it require the hypothesis that the starting Riemannian manifolds
are non Ricci flat.
