Large liquid argon detectors offer one of the best avenues for the detection of galactic weakly interacting massive particles (WIMPs) via their scattering on atomic nuclei. The liquid argon target allows exquisite discrimination between nuclear and electron recoil signals via pulse-shape discrimination of the scintillation signals. Atmospheric argon (AAr), however, has a naturally occurring radioactive isotope, 39Ar, a β emitter of cosmogenic origin. For large detectors, the atmospheric 39Ar activity poses pile-up concerns. The use of argon extracted from underground wells, deprived of 39Ar, is key to the physics potential of these experiments. The DarkSide-20k dark matter search experiment will operate a dual-phase time projection chamber with 50 tonnes of radio-pure underground argon (UAr), that was shown to be depleted of 39Ar with respect to AAr by a factor larger than 1400. Assessing the 39Ar content of the UAr during extraction is crucial for the success of DarkSide-20k, as well as for future experiments of the Global Argon Dark Matter Collaboration (GADMC). This will be carried out by the DArT in ArDM experiment, a small chamber made with extremely radio-pure materials that will be placed at the centre of the ArDM detector, in the Canfranc Underground Laboratory (LSC) in Spain. The ArDM LAr volume acts as an active veto for background radioactivity, mostly γ-rays from the ArDM detector materials and the surrounding rock. This article describes the DArT in ArDM project, including the chamber design and construction, and reviews the background required to achieve the expected performance of the detector.