A discussion of asymptotic weak and strong Poincare' charges in metric gravity is given to identify the proper Hamiltonian boundary conditions. The asymptotic part of the lapse and shift functions is put equal to their analogues on Minkowski hyperplanes. By adding Dirac's ten extra variables at spatial infinity, metric gravity is extended to incorporate Dirac's ten extra first class constraints (the new ten momenta equal to the weak Poincare' charges) and this allows its deparametrization to parametrized Minkowski theories restricted to spacelike hyperplanes. The absence of supertranslations implies: i) boundary conditions identifying the family of Christodoulou-Klainermann spacetimes; ii) the restriction of foliations to those (Wigner-Sen-Witten hypersurfaces) corresponding to Wigner's hyperplanes of Minkowski rest-frame instant form. These results are generalized to tetrad gravity in the new formulation given in gr-qc/9807072, gr-qc/9807073. The evolution in the parameter labelling the leaves of the foliation is generated by the weak ADM energy. Some comments on the quantization in a completely fixed special 3-orthogonal gauge are made.