Capillary pumping is an attractive means of liquid actuation because it is a passive mechanism, i.e., it does not rely on an external energy supply during operation. The capillary flow rate generally depends on the liquid sample viscosity and surface energy. This poses a problem for capillary-driven systems that rely on a predictable flow rate and for which the sample viscosity or surface energy are not precisely known. Here, we introduce the capillary pumping of sample liquids with a flow rate that is constant in time and independent of the sample viscosity and sample surface energy. These features are enabled by a design in which a well-characterized pump liquid is capillarily imbibed into the downstream section of the pump and thereby pulls the unknown sample liquid into the upstream pump section. The downstream pump geometry is designed to exert a Laplace pressure and fluidic resistance that are substantially larger than those exerted by the upstream pump geometry on the sample liquid. Hence, the influence of the unknown sample liquid on the flow rate is negligible. We experimentally tested pumps of the new design with a variety of sample liquids, including water, different samples of whole blood, different samples of urine, isopropanol, mineral oil, and glycerol. The capillary filling speeds of these liquids vary by more than a factor 1000 when imbibed to a standard constant cross-section glass capillary. In our new pump design, 20 filling tests involving these liquid samples with vastly different properties resulted in a constant volumetric flow rate in the range of 20.96-24.76 μL/min. We expect this novel capillary design to have immediate applications in lab-on-a-chip systems and diagnostic devices.