  We have found an effective fueling process to a central region of galaxies
with weak bar-like distortion by two dimensional hydrodynamical simulations.
Gravitational instability of an elongated gas ring at the inner Lindblad
resonance (ILR), which has been reported as an effective fueling mechanism, are
not needed for this fueling process. A massive gaseous disk in a central region
of galaxies sensitively responds to the weakly distorted potential, and a large
amount of gas can be fed into within $1/20$ of a core radius of the potential
in several $10^7$ yr. The ILRs, the dissipative nature of the gas, and
self-gravity of the gas are essential for triggering this effective fueling.
The accumulation process has not been ever known: the gas accumulates to form a
dense `linear' structure inclined at about 45 degree with respect to the bar
potential in a {\it leading sense}. We also found that a counter rotating
gaseous core can be formed as a result of the fueling. The sense of the
rotation of the core depends on a fraction of the gas mass to the background
mass. Physical mechanism of the fueling process can be understood using a
linear theory of gaseous orbits in a weak barred potential.
