  We study the inverse conductivity problem of how to reconstruct an isotropic
electrical conductivity distribution $\gamma$ in an object from static
electrical measurements on the boundary of the object. We give an exact
reconstruction algorithm for the conductivity $\gamma\in C^{1+\epsilon}(\ol
\Om)$ in the plane domain $\Omega$ from the associated Dirichlet to Neumann map
on $\partial \Om.$ Hence we improve earlier reconstruction results. The method
used relies on a well-known reduction to a first order system, for which the
$\ol\partial$-method of inverse scattering theory can be applied.
