Ionizing radiation induces a variety of DNA damages including single-strand breaks (SSBs), double-strand breaks (DSBs), abasic sites, modified sugars, and bases. Most theoretical and experimental studies have been focused on DNA strand scissions, in particular production of DNA double-strand breaks. DSBs have been proven to be a key damage at a molecular level responsible for the formation of chromosomal aberrations, leading often to cell death. We have studied the nature of DNA damage induced directly by the pulsed 46.9-nm (26.5 eV) radiation provided by an extreme ultraviolet (XUV) capillary-discharge Ne-like Ar laser (CDL). Doses up to 45 kGy were delivered with a repetition rate of 3 Hz. We studied the dependence of the yield of SSBs and DSBs of a simple model of DNA molecule (pBR322) on the CDL pulse fluence. Agarose gel electrophoresis method was used for determination of both SSB and DSB yields. The action cross sections of the single- and double-strand breaks of pBR322 plasmid DNA in solid state were determined. We observed an increase in the efficiency of strand-break induction in the supercoiled DNA as a function of laser pulse fluence. Results are compared to those acquired at synchrotron radiation facilities and other sources of extreme-ultraviolet and soft x-ray radiation.