We present the discovery of one or two extremely faint z~6 quasars in 6.5 deg^2 utilizing a unique capability of the wide-field imaging of the Subaru/Suprime-Cam. The quasar selection was made in (i-z_B) and (z_B-z_R) colors, where z_B and z_R are bandpasses with central wavelengths of 8842A and 9841A, respectively. The color selection can effectively isolate quasars at z~6 from M/L/T dwarfs without the J-band photometry down to z_R<24.0, which is 3.5 mag. deeper than SDSS. We have selected 17 promising quasar candidates. The follow-up spectroscopy for seven targets identified one apparent quasar at z=6.156 with M_1450=-23.10. We also identified one possible quasar at z=6.041 with a faint continuum of M_1450=-22.58 and a narrow Lyman-alpha emission with HWHM=427 km/s, which cannot be distinguished from Lyman-alpha emitters. We derive the quasar luminosity function at z~6 by combining our faint quasar sample with the bright quasar samples by SDSS and CFHQS. Including our data points invokes a higher number density in the faintest bin of the quasar luminosity function than the previous estimate employed. This suggests a steeper faint-end slope than lower-z, though it is yet uncertain based on a small number of spectroscopically identified faint quasars and several quasar candidates are still remain to be diagnosed. The steepening of the quasar luminosity function at the faint-end does increase the expected emission rate of the ionizing photon, however, it only changes by a factor of ~2-6. This was found to be still insufficient for the required photon budget of reionization at z~6.
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