  The $0^{+}$ states of $^{12}\mathrm{C}$ are considered within the framework
of the microscopic three-$\alpha$-cluster model. The main attention is paid to
accurate calculation of the width of the extremely narrow near-threshold
$0^+_2$ state which plays a key role in stellar nucleosynthesis. It is shown
that the $0^{+}_2$-state decays by means of the sequential mechanism
${^{12}\mathrm{C}} \to \alpha+{^8\mathrm{Be}} \to 3\alpha$. Calculations are
performed for a number of effective $\alpha - \alpha$ potentials which are
chosen to reproduce both energy and width of $^8\mathrm{Be}$. The parameters of
the additional three-body potential are chosen to fix both the ground and
excited state energies at the experimental values. The dependence of the width
on the parameters of the effective $\alpha - \alpha$ potential is studied in
order to impose restrictions on the potentials.
