  We use an underground counting lab with an extremely low background to
perform an activity measurement for the $^{12}$C+$^{13}$C system with energies
down to $E\rm_{c.m.}$=2.323 MeV, at which the $^{12}$C($^{13}$C,$p$)$^{24}$Na
cross section is found to be 0.22(7) nb. The $^{12}$C+$^{13}$C fusion cross
section is derived with a statistical model calibrated using experimental data.
Our new result of the $^{12}$C+$^{13}$C fusion cross section is the first
decisive evidence in the carbon isotope systems which rules out the existence
of the astrophysical S-factor maximum predicted by the phenomenological
hindrance model, while confirming the rising trend of the S-factor towards
lower energies predicted by other models, such as CC-M3Y+Rep, DC-TDHF, KNS, SPP
and ESW. After normalizing the model predictions with our data, a more reliable
upper limit is established for the $^{12}$C+$^{12}$C fusion cross sections at
stellar energies.
