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Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p\>0$, and let $K^{+}$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^{+}$-schemes $X=Spec\left(R\right)$ with good or semi-stable reduction to the Galois cohomology of the fundamental group ${\pi }_1\left(X_{\overline{K}}\right)$ of the geometric generic fibre with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings’ theory of almost étale extensions.