The Bruce nuclear site in Canada has been proposed to host a Deep Geologic Repository (DGR) for Low and Intermediate Level Radioactive Waste (L&ILW). The repository would be constructed within a low permeability, argillaceous limestone, the Upper Ordovician age Cobourg Formation. Here, we present the results of two steady-state laboratory hydraulic conductivity tests performed to measure the intrinsic permeability of rock core samples from the Cobourg and overlying Queenston shale formations; both samples were measured under an isotropic confining pressure using a constant head approach. Pump pressures and volumes were recorded for upstream and downstream pumps, throughout testing. The resulting hydraulic inflow and outflow rates were measured for each sample under two different pressure gradients, yielding exceptionally low values of permeability (on the order of 10-22 m2 or 0.1 nD). These data provide further evidence of the applicability of existing steady-state experimental methods to obtain reliable estimates of extremely low permeabilities from rock core samples under re-established in-situ stress conditions. The exceptionally low permeability of these formations, consistent with in-situ testing and formation scale estimates obtained during the site characterisation program, along with their low porosities, renders them an effective barrier to hydraulic flow for the purpose of geological isolation.