  Consider discrete conformal maps defined on the basis of two conformally
equivalent triangle meshes, that is edge lengths are related by scale factors
associated to the vertices. Given a smooth conformal map $f$, we show that it
can be approximated by such discrete conformal maps $f^\epsilon$. In
particular, let $T$ be an infinite regular triangulation of the plane with
congruent triangles and only acute angles (i.e.\ $<\pi/2$). We scale this
tiling by $\epsilon>0$ and approximate a compact subset of the domain of $f$
with a portion of it. For $\epsilon$ small enough we prove that there exists a
conformally equivalent triangle mesh whose scale factors are given by
$\log|f'|$ on the boundary. Furthermore we show that the corresponding discrete
conformal maps $f^\epsilon$ converge to $f$ uniformly in $C^1$ with error of
order $\epsilon$.
