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Inductively Coupled Plasma Mass Spectrometry (ICP-MS) allows for rapid, high-sensitivity determination of trace impurities, notably the primordial radioisotopes $^{238}$U and $^{232}$Th, in candidate materials for low-background rare-event search experiments. We describe the setup and characterisation of a dedicated low-background screening facility at University College London where we operate an Agilent 7900 ICP-MS. The impact of reagent and carrier gas purity is evaluated and we show that twice-distilled ROMIL-SpA-grade nitric acid and zero-grade Ar gas delivers similar sensitivity to ROMIL-UpA-grade acid and research grade gas. A straightforward procedure for sample digestion and analysis of materials with U/Th concentrations down to 10 ppt g/g is presented. This includes the use of $^{233}$U and $^{230}$Th spikes to correct for signal loss from a range of sources and verification of $^{238}$U and $^{232}$Th recovery through digestion and analysis of a certified reference material with a complex sample matrix. Finally, we demonstrate assays and present results from two sample preparation and assay methods: a high-sensitivity measurement of ultra-pure Ti using open digestion techniques, and a closed vessel microwave digestion of a nickel-chromium-alloy using a multi-acid mixture.