  In this paper, we address a classical case of the Calder\'on (or
conductivity) inverse problem in dimension two. We aim to recover the location
and the shape of a single cavity $\omega$ (with boundary $\gamma$) contained in
a domain $\Omega$ (with boundary $\Gamma$) from the knowledge of the
Dirichlet-to-Neumann (DtN) map $\Lambda_\gamma: f \longmapsto \partial_n
u^f|_{\Gamma}$, where $u^f$ is harmonic in $\Omega\setminus\overline{\omega}$,
$u^f|_{\Gamma}=f$ and $u^f|_{\gamma}=c^f$, $c^f$ being the constant such that
$\int_{\gamma}\partial_n u^f\,{\rm d}s=0$. We obtain an explicit formula for
the complex coefficients $a_m$ arising in the expression of the Riemann map
$z\longmapsto a_1 z + a_0 + \sum_{m\leqslant -1} a_m z^{m}$ that conformally
maps the exterior of the unit disk onto the exterior of $\omega$. This formula
is derived by using two ingredients: a new factorization result of the DtN map
and the so-called generalized P\'olia-Szeg\"o tensors (GPST) of the cavity. As
a byproduct of our analysis, we also prove the analytic dependence of the
coefficients $a_m$ with respect to the DtN. Numerical results are provided to
illustrate the efficiency and simplicity of the method.
