The radioactive noble gas radon can be a serious background source in the underground particle physics experiments studying processes that deposit energy comparable to its decay products. Low energy solar neutrino measurements at Super-Kamiokande suffer from these backgrounds and therefore require precise characterization of the radon concentration in the detectors ultra-pure water. For this purpose, we have developed a measurement system consisting of a radon extraction column, a charcoal trap, and a radon detector. In this article we discuss the design, calibration, and performance of the radon extraction column. We also describe the design of the measurement system and evaluate its performance, including its background. Using this system we measured the radon concentration in Super-Kamiokandes water between May 2014 and October 2015. The measured radon concentration in the supply lines of the water circulation system was $1.74pm0.14~mathrm{mBq/m^{3}}$ and in the return line was $9.06pm0.58~mathrm{mBq/m^{3}}$. Water sampled from the center region of the detector itself had a concentration of $<0.23~mathrm{mBq/m^{3}}$ ($95%$ C.L.) and water sampled from the bottom region of the detector had a concentration of $2.63pm0.22~mathrm{mBq/m^{3}}$.
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