We consider the inverse conductivity problem with one measurement for the equation $div((\sigma_1+(\sigma_2-\sigma_1)\chi_D)\nabla{u})=0$ determining the unknown inclusion $D$ included in $\Omega$. We suppose that $\Omega$ is the unit disk of $\mathbb{R}^2$. With the tools of the conformal mappings, of elementary Fourier analysis and also the action of some quasi-conformal mapping on the Sobolev space $\sH^{1/2}(S^1)$, we show how to approximate the Dirichlet-to-Neumann map when the original inclusion $D$ is a $\varepsilon-$ approximation of a disk. This enables us to give some uniqueness and stability results.
