We developed a CANDLES-III system to study the neutrino-less double beta (0$ ubetabeta$) decay of $^{48}$Ca. The proposed system employs 96 CaF$_{2}$ scintillation crystals (305 kg) with natural Ca ($^{rm nat.}$Ca) isotope which corresponds 350,g of $^{48}$Ca. External backgrounds were rejected using a 4$pi$ active shield of a liquid scintillator surrounding the CaF$_2$ crystals. The internal backgrounds caused by the radioactive impurities within the CaF$_2$ crystals can be reduced effectively through analysis of the signal pulse shape. We analyzed the data obtained in the Kamioka underground for a live-time of 130.4,days to evaluate the feasibility of the low background measurement with the CANDLES-III detector. Using Monte Carlo simulations, we estimated the background rate from the radioactive impurities in the CaF$_{2}$ crystals and the rate of high energy $gamma$-rays caused by the (n, $gamma$) reactions induced by environmental neutrons. The expected background rate was in a good agreement with the measured rate, i.e., approximately 10$^{-3}$ events/keV/yr/(kg of $^{rm nat.}$Ca), in the 0$ ubetabeta$ window. In conclusion, the background candidates were estimated properly by comparing the measured energy spectrum with the background simulations. With this measurement method, we performed the first search for 0$ ubetabeta$ decay in a low background condition using a detector with a Ca isotope, in which the Ca present was not enriched, in a scale of hundreds of kg. The $^{48}$Ca isotope has a high potential for use in 0$ ubetabeta$ decay search, and is expected to be useful for the development of a next-generation detector for highly sensitive measurements.
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