Rotationally resolved infrared spectra of Mg+-H-2 and Mg+-D-2 are recorded in the H-H (4025-4080 cm(-1)) and D-D (2895-2945 cm(-1)) stretch regions by monitoring Mg+ photofragments. The v(HH) and v(DD) transitions of Mg+-H-2 and Mg+-D-2 are red-shifted by 106.2 +/- 1.5 and 76.0 +/- 0.1 cm(-1) respectively from the fundamental vibrational transitions of the free H-2 and D-2 molecules. The spectra are consistent with a T-shaped equilibrium structure in which the Mg+ ion interacts with a slightly perturbed H-2 or D-2 molecule. From the spectroscopic constants, a vibrationally averaged intermolecular separation of 2.716 angstrom (2.687 angstrom) is deduced for the ground state of Mg+-H-2 (Mg+-D-2), decreasing by 0.037 angstrom (0.026 angstrom) when the H2 (D2) subunit is vibrationally excited.