One of the most troublesome contributions to the NNLO QCD corrections to
${\bar B}\to X_s\gamma$ originates from three-loop matrix elements of
four-quark operators. A part of this contribution that is proportional to the
QCD beta-function coefficient $\beta_0$ was found in 2003 as an expansion in
$m_c/m_b$. In the present paper, we evaluate the asymptotic behaviour of the
complete contribution for $m_c \gg m_b/2$. The asymptotic form of the
$\beta_0$-part matches the small-$m_c$ expansion very well at the threshold
$m_c = m_b/2$. For the remaining part, we perform an interpolation down to
the measured value of $m_c$, assuming that the $\beta_0$-part is a good
approximation at $m_c=0$. Combining our results with other contributions to
the NNLO QCD corrections, we find ${\cal B}({\bar B}\to X_s\gamma) = (3.15 \pm
0.23) \times 10^{-4}$ for $E_{\gamma} > 1.6\;$GeV in the ${\bar B}$-meson rest
frame. The indicated error has been obtained by adding in quadrature
the following uncertainties: non-perturbative (5\%), parametric (3\%),
higher-order %(${\cal O}(\alpha_s^3)$)
perturbative (3\%), and the interpolation ambiguity (3\%).