Thin-film stacks | consisting of a ferromagnetic-metal layer and a heavymetal layer are spintronic model systems. Here, we present a method to measure the ultrabroadband spin conductance across a layer between and at 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 G s has 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 d to 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.