  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.
