  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 $\alpha$ and $\beta$
particles. Uranium and thorium contamination in the CCD bulk was measured
through $\alpha$ 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 $\beta$ decays. The decay rate of $^{32}$Si
was found to be $80^{+110}_{-65}$ kg$^{-1}$ d$^{-1}$ (95% CI). An upper limit
of $\sim$35 kg$^{-1}$ d$^{-1}$ (95% CL) on the $^{210}$Pb decay rate was
obtained independently by $\alpha$ spectroscopy and the $\beta$ decay sequence
search. These levels of radioactive contamination are sufficiently low for the
successful operation of CCDs in the forthcoming 100 g DAMIC detector.
