A rational map from a normal variety to a proper variety extends in codimension one

By [F1], $R={\mathcal{O}}_{X,\eta }$ is a DVR with fraction field $k(X)$. Apply [F2] to extend the generic morphism $\mathrm{Spec}k(X)\to Y$ uniquely to $\mathrm{Spec}R\to Y$. Choose an affine open of $Y$ containing the image of the closed point of this local spectrum; its inverse image contains that closed point and hence is the whole local spectrum.

The chosen target affine ring is finitely generated over $k$. The images of its finitely many generators in $R$ are regular on a common open neighbourhood of $\eta$ in $X$. Its relations hold there because they hold in the function field of the integral $X$. These elements therefore give a morphism on that neighbourhood agreeing with the rational map generically. Separatedness of $Y$ glues such representatives, so $\eta$ belongs to the maximal domain. No codimension-one point can occur in its complement; the generic point was already in the domain. AC is inherited from [F1]–[F2].