A well-defined cluster containing 12 equivalent platinum atoms was prepared by ion exchange of an NaY zeolite, followed by hydrogen reduction. It was characterized by electron paramagnetic resonance (EPR) spectroscopy, hyperfine sublevel correlation (HYSCORE), and theoretical calculations. Combing the results of the experiments with density functional calculations, the likely structure of this cluster is icosahedral Pt13Hm, possibly with a low positive charge. The adsorbed H/D on the Pt cluster surface can be exchanged reversibly at room temperature. From H/D desorption experiments, an H2 binding energy of 1.36 eV is derived, in reasonable agreement with the calculated value but clearly larger than that for a (111) Pt single-crystal surface, revealing a finite size effect. While the hydrogen-covered cluster should clearly be regarded as a molecule, it is conceivable that the cluster adopts metallic character upon hydrogen desorption. It is likely that up to m=30 H atoms bind to this cluster with 12 surface atoms, which has important implications for the determination of the dispersion of small Pt catalyst particles by hydrogen chemisorption. Calculations as well as experiments give evidence of an interesting magnetic behavior with high-spin states playing a prominent role. There are strong indications that a reservoir of EPR silent but structurally similar clusters exists which can partly be converted to EPR visible species by H/D exchange or by gas adsorption.