  We present measurements of bulk radiocontaminants in the high-resistivity
silicon CCDs from the DAMIC at SNOLAB experiment. We utilize the exquisite
spatial resolution of CCDs to discriminate between $\alpha$ and $\beta$ decays,
and to search with high efficiency for the spatially-correlated decays of
various radioisotope sequences. Using spatially-correlated $\beta$ decays, we
measure a bulk radioactive contamination of $^{32}$Si in the CCDs of $140 \pm
30$ $\mu$Bq/kg, and place an upper limit on bulk $^{210}$Pb of $<
160~\mu$Bq/kg. Using similar analyses of spatially-correlated bulk $\alpha$
decays, we set limits of $< 11$ $\mu$Bq/kg (0.9 ppt) on $^{238}$U and of $<
7.3$ $\mu$Bq/kg (1.8 ppt) on $^{232}$Th. The ability of DAMIC CCDs to identify
and reject spatially-coincident backgrounds, particularly from $^{32}$Si, has
significant implications for the next generation of silicon-based dark matter
experiments, where $\beta$'s from $^{32}$Si decay will likely be a dominant
background. This capability demonstrates the readiness of the CCD technology to
achieve kg-scale dark matter sensitivity.
