We established a method to assay $^{210}$Pb and $^{210}$Po contaminations in the bulk of copper samples using a low-background alpha particle counter. The achieved sensitivity for the $^{210}$Pb and $^{210}$Po contaminations reaches a few mBq/kg. Due to this high sensitivity, the $^{210}$Pb and $^{210}$Po contaminations in oxygen free copper bulk were identified and measured for the first time. The $^{210}$Pb contaminations of our oxygen free copper samples were 17-40 mBq/kg. Based on our investigation of copper samples in each production step, the $^{210}$Pb in oxygen free copper was understood to be a small residual of an electrolysis process. This method to measure bulk contaminations of $^{210}$Pb and $^{210}$Po could be applied to other materials.
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