In this paper we prove the breakdown of an heteroclinic connection inthe analytic versal unfoldings of the generic Hopf-Zero singularity. Thisheteroclinic orbit appears at any order if one performs the normal formaround the origin, therefore it is a phenomenon \beyond all orders". Inthis paper we provide a formula for the distance between the correspond-ing stable and unstable one dimensional manifolds which is given by anexponentially small function in the perturbation parameter. Our resultapplies both for conservative unfoldings and for dissipative ones in someopen regions of the parameter plane