We studied the relative stability and atomic structure of five $C_{20}^{2+}$ isomers obtained by two-electron ionization of a $C_{20}$ cage (the smallest fullerene). All the isomers are bond-stretch isomers, i.e., they differ in bond length. In particular, in one of the isomers with Ih symmetry, all the bond lengths are equal. Full geometry optimizations of the dipositive ion $C_{20}^{2+}$ were performed using the hybrid density functional (B3LYP/6-31G(d)) methods. All isomers were found to be true minima by frequency analysis at the level of B3LYP/6-31G(d) under the reinforced tight convergence criterion and a pruned (99,590) grid. The zero-point correction energy for the cage bond-stretch isomers was in the increasing order $D_{2h}