We present measurements of radioactive contamination in the high-resistivity silicon charge-coupled devices (CCDs) used by the DAMIC experiment to search for dark matter particles. Novel analysis methods, which exploit the unique spatial resolution of CCDs, were developed to identify α and β particles. Uranium and thorium contamination in the CCD bulk was measured through α spectroscopy, with an upper limit on the (238)U ((232)Th) decay rate of 5 (15) kg(−)(1) d(−)(1) at 95% CL. We also searched for pairs of spatially correlated electron tracks separated in time by up to tens of days, as expected from (32)Si –(32)P or (210)Pb –(210)Bi sequences of β decays. The decay rate of (32)Si was found to be 80(+110)(−)(65) kg(−)(1) d(−)(1) (95% CI). An upper limit of ~35 kg(−)(1) d(−)(1) (95% CL) on the (210)Pb decay rate was obtained independently by α spectroscopy and the β decay sequence search. These levels of radioactive contamination are sufficiently low for the successful operation of CCDs in the forthcoming 100 g DAMIC detector.