  Stellar carbon synthesis occurs exclusively via the $3\alpha$ process, in
which three $\alpha$ particles fuse to form $^{12}$C in the excited Hoyle
state, followed by electromagnetic decay to the ground state. The Hoyle state
is above the $\alpha$ threshold, and the rate of stellar carbon production
depends on the radiative width of this state. The radiative width cannot be
measured directly, and must instead be deduced by combining three separately
measured quantities. One of these quantities is the $E0$ decay branching ratio
of the Hoyle state, and the current $10$\% uncertainty on the radiative width
stems mainly from the uncertainty on this ratio. The $E0$ branching ratio was
deduced from a series of pair conversion measurements of the $E0$ and $E2$
transitions depopulating the $0^+_2$ Hoyle state and $2^+_1$ state in $^{12}$C,
respectively. The excited states were populated by the $^{12}$C$(p,p^\prime)$
reaction at 10.5 MeV beam energy, and the pairs were detected with the
electron-positron pair spectrometer, Super-e, at the Australian National
University. The deduced branching ratio required knowledge of the proton
population of the two states, as well as the alignment of the $2^+_1$ state in
the reaction. For this purpose, proton scattering and $\gamma$-ray angular
distribution experiments were also performed. An $E0$ branching ratio of
$\Gamma^{E0}_{\pi}/\Gamma=8.2(5)\times10^{-6}$ was deduced in the current work,
and an adopted value of $\Gamma^{E0}_{\pi}/\Gamma=7.6(4)\times10^{-6}$ is
recommended based on a weighted average of previous literature values and the
new result. The new recommended value for the $E0$ branching ratio is about 14%
larger than the previous adopted value of
$\Gamma^{E0}_{\pi}/\Gamma=6.7(6)\times10^{-6}$, while the uncertainty has been
reduced from 9% to 5%.
