In this paper we investigate the hyperspace GH(Rn) of the isometry classes of all non-empty compact subsets of a Euclidean space Rn in the Gromov-Hausdorff metric. It is continuation of our previous paper 5, where it was established that for any n≥1, GH(Rn) is homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of Rn equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This model is further applied to determine the complete topological structure of GH(Rn) for all n≥1, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n≥1, GH(Bn) is homeomorphic to the Hilbert cube, where Bn denotes the closed unit ball of Rn.