  Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and
$\hat \sigma$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is
given, then starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat
\sigma}$ on $\partial X$, we give an explicit procedure to find a unique
Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity
$\sigma$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F:
X \to Y$ which transforms $\hat \sigma$ into $\sigma$.
  As a corollary we obtain the following uniqueness result: if $\sigma_1,
\sigma_2$ are two smooth anisotropic conductivities on $X$ with
$\Lambda_{\sigma_1}= \Lambda_{\sigma_2}$, then there exists a smooth
diffeomorphism $\Phi: \bar X \to \bar X$ which transforms $\sigma_1$ into
$\sigma_2$.
