The Russian Emission Detector 100 (RED-100) under construction at the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) is designed to detect the presently undiscovered effect of coherent neutrino scattering. One of the factors limiting the sensitivity of the detector is the spontaneous decay of uranium and thorium in the detector materials. Radioactive impurities in detector materials at levels of parts per billion can significantly affect the sensitivity. Five random samples of titanium and one of copper from materials used in the construction of the detector were selected for assay. The concentration of (232)Th and (238)U were measured by inductively coupled plasma mass spectrometry (ICP- MS) in solid titanium using both: solutions in acids and direct sampling by laser ablation (LA-ICP-MS). The LA- ICP-MS method allowed us to determine (238)U and (232)Th at subnanogram per gram levels. This method is much faster than ICP-MS with liquid injection. The titanium samples studied have impurities in the range between 1 ng g(-1) and 21 ng g(-1) for (238)U and 3 ng g(-1) and 31 ng g(-1) for (232)Th. In copper we set upper limits of 0.4 ng g(-1) for (238)U and 1 ng g(-1)for (232)Th. The total activity of the cryostat constructed from materials studied was estimated to be 43 Bq.