Use of the tritopic bridging ligand 1,3,5-benzenetristetrazolate (BTT3-) enables formation of [Mn(DMF)6]3[(Mn4Cl)3(BTT)8(H2O)12]2.42DMF.11H2O.20CH3OH, featuring a porous metal-organic framework with a previously unknown cubic topology. Crystals of the compound remain intact upon desolvation and show a total H2 uptake of 6.9 wt % at 77 K and 90 bar, which at 60 g H2/L provides a storage density 85% of that of liquid hydrogen. The material exhibits a maximum isosteric heat of adsorption of 10.1 kJ/mol, the highest yet observed for a metal-organic framework. Neutron powder diffraction data demonstrate that this is directly related to H2 binding at coordinatively unsaturated Mn2+ centers within the framework.