While the pure elements tend to exhibit Type-I rather than Type-II superconductivity, nearly all compound superconductors are Type-II, with only a few known exceptions. We report single crystal growth and physical characterization of the rhenium aluminide Al6Re, which we conclude is a Type-I superconductor based on magnetization, ac-susceptibility, and specific-heat measurements. This detection of superconductivity, despite the strong similarity of Al6Re to a family of W and Mo aluminides that do not superconduct, suggests that these aluminides are an ideal testbed for identifying the relative importance of valence electron count and inversion symmetry in determining whether a material will superconduct.