Diamine-appended Mg₂(dobpdc) (dobpdc^4- = 4,4'-dioxidobiphenyl-3,3'-dicarboxylate) metal-organic frameworks are promising candidates for carbon capture that exhibit exceptional selectivities and high capacities for CO₂. To date, CO₂ uptake in these materials has been shown to occur predominantly via a chemisorption mechanism involving CO₂ insertion at the amine-appended metal sites, a mechanism that limits the capacity of the material to ∼1 equiv of CO₂ per diamine. Herein, we report a new framework, pip2-Mg₂(dobpdc) (pip2 = 1-(2-aminoethyl)piperidine), that exhibits two-step CO₂ uptake and achieves an unusually high CO₂ capacity approaching 1.5 CO₂ per diamine at saturation. Analysis of variable-pressure CO₂ uptake in the material using solid-state nuclear magnetic resonance (NMR) spectroscopy and in situ diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS) reveals that pip2-Mg₂(dobpdc) captures CO₂ via an unprecedented mechanism involving the initial insertion of CO₂ to form ammonium carbamate chains at half of the sites in the material, followed by tandem cooperative chemisorption and physisorption. Powder X-ray diffraction analysis, supported by van der Waals-corrected density functional theory, reveals that physisorbed CO₂ occupies a pocket formed by adjacent ammonium carbamate chains and the linker. Based on breakthrough and extended cycling experiments, pip2-Mg₂(dobpdc) exhibits exceptional performance for CO₂ capture under conditions relevant to the separation of CO₂ from landfill gas. More broadly, these results highlight new opportunities for the fundamental design of diamine-Mg₂(dobpdc) materials with even higher capacities than those predicted based on CO₂ chemisorption alone.