  In our previous paper, we have shown that a gas disk in the nuclear region of
a barred galaxy which contains a central supermassive black hole (SMBH) rapidly
evolves into a nuclear gas ring by the effect of an additional inner Lindblad
resonance caused by the SMBH. In this paper, we investigate the fate of the gas
ring, involving self-gravity of gas, using two-dimensional hydrodynamical
simulations. We find that the gas ring becomes gravitationally unstable for a
gas surface density of gas above a critical value, and fragments into several
gas clumps. Some denser clumps increase their mass via the accretion of the
surrounding gas and collisions with other clumps, and finally a very massive
gas clump (10^7 M_sun) is formed. Due to the torque from the most massive
clump, a part of the gas in the ring loses its angular momentum and falls into
the galactic center. As a result, a nuclear gas disk (50 pc) is formed around
the SMBH. The accretion rate for $R<50$ pc attains about 1 M_sun/yr for
3.5*10^7 yr. At the final phase of the bar-driven fueling, self-gravity is
crucial for the angular momentum transfer of the gas. This is a new mechanism
for gas fueling to the vicinity of the SMBH.
