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Thin-film stacks F|Hconsisting of a ferromagnetic-metal layer Fand a heavy-metal layer Hare spintronic model systems. Here, we present a method to measure the ultrabroadband spin conductance across a layer Xbetween Fand Hat terahertz frequencies, which are the natural frequencies of spin-transport dynamics. We apply our approach to MgO tunneling barriers with thickness d= 0-6 Å. In the time domain, the spin conductance Gshas two components. An instantaneous feature arises from processes like coherent spin tunneling. Remarkably, a longer-lived component is a hallmark of incoherent resonant spin tunneling mediated by MgO defect states, because its relaxation time grows monotonically with dto as much as 270 fs at d= 6.0 Å. Our results are in full agreement with an analytical model. They indicate that terahertz spin-conductance spectroscopy will yield new and relevant insights into ultrafast spin transport in a wide range of spintronic nanostructures.