CRISPR-Cas (clustered regularly interspaced short palindromic repeats-CRISPR-associated) systems are RNA-guided adaptive immunity pathways used by bacteria and archaea to defend against phages and plasmids. Type III-A systems use a multisubunit interference complex called Csm, containing Cas proteins and a CRISPR RNA (crRNA) to target cognate nucleic acids. The Csm complex is intriguing in that it mediates RNA-guided targeting of both RNA and transcriptionally active DNA, but the mechanism is not well understood. Here, we overexpressed the five components of the Thermus thermophilus (T. thermophilus) Type III-A Csm complex (TthCsm) with a defined crRNA sequence, and purified intact TthCsm complexes from E. coli cells. The complexes were thermophilic, targeting complementary ssRNA more efficiently at 65°C than at 37°C. Sequence-independent, endonucleolytic cleavage of single-stranded DNA (ssDNA) by TthCsm was triggered by recognition of a complementary ssRNA, and required a lack of complementarity between the first 8 nucleotides (5' tag) of the crRNA and the 3' flanking region of the ssRNA. Mutation of the histidine-aspartate (HD) nuclease domain of the TthCsm subunit, Cas10/Csm1, abolished DNA cleavage. Activation of DNA cleavage was dependent on RNA binding but not cleavage. This leads to a model in which binding of an ssRNA target to the Csm complex would stimulate cleavage of exposed ssDNA in the cell, such as could occur when the RNA polymerase unwinds double-stranded DNA (dsDNA) during transcription. Our findings establish an amenable, thermostable system for more in-depth investigation of the targeting mechanism using structural biology methods, such as cryo-electron microscopy and x-ray crystallography.