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Inductively coupled plasma mass (ICP-MS) spectroscopy is widely used for screening materials of low background detectors in dark matter and double beta decay searches due to its high sensitivity to trace $^{238}$U and $^{232}$Th. This work describes a novel co-precipitation approach to measure the amount of $^{238}$U in high-purity copper to sub-ppt level. Such an approach allows the pre-concentration of U and removal of the matrix, by selecting a proper precipitator to co-precipitate with $^{238}$U and using excess ammonia water to separate the uranium hydroxide from copper by forming water-soluble tetra-amminecopper (II). The isotope dilution method and standard addition method were both used to mitigate the matrix effect and cross-check each other. The latter was also used to measure the recovery efficiency of $^{238}$U by using $^{233}$U as the tracer. The method detection limit (MDL) reached $\sim$0.1 pg $^{238}$U /g Cu for both methods while the recovery efficiency of uranium robustly remains 65\%--85\%. Various sources of interference in the ICP-MS analysis were thoroughly evaluated, and the contamination from reagents were found to be the dominant factor that affected the MDL. Further purification will allow significant improvements in the MDL. This co-precipitate approach can be easily extended to measure $^{232}$Th by using $^{229}$Th as the tracer.