Type I CRISPR-Cas loci provide prokaryotes with a nucleic-acid-based adaptive immunity against foreign DNA. Immunity involves adaptation, the integration of ~30-bp DNA fragments, termed prespacers, into the CRISPR array as spacers, and interference, the targeted degradation of DNA containing a protospacer. Interference-driven DNA degradation can be coupled with primed adaptation, in which spacers are acquired from DNA surrounding the targeted protospacer. Here we develop a method for strand-specific, high-throughput sequencing of DNA fragments, FragSeq, and apply this method to identify DNA fragments accumulated in Escherichia coli cells undergoing robust primed adaptation by a type I-E or type I-F CRISPR-Cas system. The detected fragments have sequences matching spacers acquired during primed adaptation and function as spacer precursors when introduced exogenously into cells by transformation. The identified prespacers contain a characteristic asymmetrical structure that we propose is a key determinant of integration into the CRISPR array in an orientation that confers immunity.