  The radiative capture cross sections of $^{12}$C($\alpha$,$\gamma$)$^{16}$O
and derived reaction rates are calculated from the direct capture potential
model. The resulting $S$-factor at low energies is found to be dominated by
$E$2 transition to the $^{16}$O ground state. The $E$1 and $E$2 $S$-factors at
$E_{c.m.}=0.3$ MeV are $S_{E1}\approx3$ keV~b and $S_{E2}=150^{+41}_{-17}$
keV~b, respectively. The sum of the cascade transition through the excited
state of $^{16}$O is $S_{\rm casc}= 18\pm4.5$ keV~b. The derived reaction rates
at low temperatures seem to be concordant with those from the previous
evaluation. For astrophysical applications, our reaction rates below $T_9=3$
are provided in an analytic expression.
