Although numerous doubly positively charged diatomic molecules (diatomic dications) are known from investigations using mass spectrometry and ab initio quantum chemistry, only three of them, NO²⁺, N₂²⁺ and DCl²⁺, have been studied using rotationally resolved optical spectroscopy and only about a dozen by vibrationally resolved double-ionization methods. So far, no thermodynamically stable diatomic dication has been characterized spectroscopically, primarily because of experimental difficulties associated with their synthesis in sufficient densities in the gas phase. Indeed, such molecules typically involve, as constituents, rare-gas, halogen, chalcogen, and metal atoms. We report here on a new approach to characterize molecular dications based on high-resolution photoelectron spectroscopy of the singly charged parent molecular cation and present the first spectroscopic characterization of a thermodynamically stable diatomic dication, MgAr²⁺. From the fully resolved vibrational and partially resolved rotational structures of the photoelectron spectra of ²⁴MgAr⁺ and ²⁶MgAr⁺, we determined the potential-energy function of the electronic ground state of MgAr²⁺, its dissociation (binding) energy (D₀ = 10 690(3) cm⁻¹), and its harmonic (ω_e(²⁴MgAr²⁺) = 327.02(11) cm⁻¹) and anharmonic (ω_ex_e(²⁴MgAr²⁺) = 2.477(15) cm⁻¹) vibrational constants. The analysis enables us to explain quantitatively how the strong bond arises in this dication despite the fact that Ar and Mg²⁺ both have a full-shell rare-gas electronic configuration.