Purpose: To quantify the relative biological effectiveness (RBE) of ultra-high dose rate (FLASH) irradiation relative to conventional dose rate irradiation in plasmid DNA to elucidate the DNA-damaging effects of FLASH in the absence of supervenient effects (e.g., immune/inflammatory responses). Methods: We irradiated six sets of plasmid DNA-based detectors at nominal dose rates of 10 Gy/min, 50 Gy/s, or 100 Gy/s, with each set receiving doses of either ~20 Gy or ~30 Gy. Afterwards, we applied agarose gel electrophoresis to sort the DNA into localized bands associated with single strand breaks (SSBs), double strand breaks (DSBs), and undamaged DNA. We then performed a quantitative analysis of the gel images to calculate the relative yields of SSBs and DSBs. A mathematical model was used to compute the relative rates of SSBs and DSBs, which allowed us to plot the relative yields of this damage as a function of dose. We chose a biological endpoint of a 2% DSB yield to calculate an RBE of the FLASH dose rate relative to the standard dose rate beam. Results: The RBEs of FLASH beams at dose rates of 50 Gy/s and 100 Gy/s relative to conventional beams at dose rates of 10 Gy/min were 0.53 ± 0.30 and 0.55 ± 0.35, respectively. Conclusion: The FLASH effect is apparent in plasmid DNA, suggesting that the DNA damage-sparing mechanisms that underlie it are present even in the absence of any higher-level biological processes.