  The uncertainty of the theoretical prediction of the $\bar B \to X_s \gamma$
branching ratio at NLL level is dominated by the charm mass renormalization
scheme ambiguity. In this paper we calculate those NNLL terms which are related
to the renormalization of $m_c$, in order to get an estimate of the
corresponding uncertainty at the NNLL level. We find that these terms
significantly reduce (by typically a factor of two) the error on ${BR}(\bar B
\to X_s \gamma)$ induced by the definition of $m_c$. Taking into account the
experimental accuracy of around 10% and the future prospects of the $B$
factories, we conclude that a NNLL calculation would increase the sensitivity
of the observable $\bar B \to X_s \gamma$ to possible new degrees of freedom
beyond the SM significantly.
