  We present the discovery of six new quasars at z~6 selected from the Sloan
Digital Sky Survey (SDSS) southern survey, a deep imaging survey obtained by
repeatedly scanning a stripe along the celestial equator. The six quasars are
about two magnitudes fainter than the luminous z~6 quasars found in the SDSS
main survey and one magnitude fainter than the quasars reported in Paper I
(Jiang et al. 2008). Four of them comprise a complete flux-limited sample at
21<z_AB<21.8 over an effective area of 195 deg^2. The other two quasars are
fainter than z_AB=22 and are not part of the complete sample. The quasar
luminosity function at z~6 is well described as a single power law
\Phi(L_{1450}) \propto L_{1450}^{\beta} over the luminosity range
-28<M_{1450}<-25. The best-fitting slope \beta varies from -2.6 to -3.1,
depending on the quasar samples used, with a statistical error of 0.3-0.4.
About 40% of the quasars discovered in the SDSS southern survey have very
narrow Lya emission lines, which may indicate small black hole masses and high
Eddington luminosity ratios, and therefore short black hole growth time scales
for these faint quasars at early epochs.
