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Asymptotic normalization coefficients (ANC) determine the overall normalization of cross sections of peripheral radiative capture reactions. In the present paper, we treat the ANC $C$ for the virtual decay $^{16}$O$(0^+; 6.05$ MeV)$\to α+^{12}$C(g.s.), the known values of which are characterized by a large spread $(0.29-1.65)\times 10^3$ fm$^{-1/2}$. The ANC $C$ is found by analytic continuation in the energy of the $α^{12}$C $s$-wave scattering amplitude, known from the phase-shift analysis of experimental data, to the pole corresponding to the $^{16}$O bound state and lying in the unphysical region of negative energies. To determine $C$, two different methods of analytic continuation are used. In the first method, the scattering data are approximated by the sum of polynomials in energy in the physical region and then extrapolated to the pole. The best way of extrapolation is chosen on the basis of the exactly solvable model. Within the second approach, the ANC $C$ is found by solving the Schrödinger equation for the two-body $α^{12}$C potential, the parameters of which are selected from the requirement of the best description of the phase-shift analysis data at a fixed experimental binding energy of $^{16}$O$(0^+; 6.05$ MeV) in the $α+^{12}$C channel. The values of the ANC $C$ obtained within these two methods lie in the interval (886--1139) fm$^{-1/2}$.