The classical KAM theorem establishes the persistence of invariant Lagrangean tori in nearly integrable Hamiltonian systems. These tori are quasi-periodic with Diophantine frequency vectors and their union is a nowhere dense set of positive measure in phase space. A long-standing question is to what extent the perturbed tori are unique. Using the fact that at the level of tori there exists a Whitney smooth conjugacy between the integrable approximation and its perturbation, we are able to prove this unicity. The unicity result is valid on a closed subset of the Diophantine torus union of full measure.