This paper proposes a new fast and stable algorithm for the reconstruction of the plasma boundary from discrete magnetic measurements taken at several locations surrounding the vacuum vessel. The resolution of this inverse problem takes two steps. In the first one we transform the set of measurements into Cauchy conditions on a fixed contour $\Gamma_O$ close to the measurement points. This is done by least square fitting a truncated series of toroidal harmonic functions to the measurements. The second step consists in solving a Cauchy problem for the elliptic equation satisfied by the flux in the vacuum and for the overdetermined boundary conditions on $\Gamma_O$ previously obtained with the help of toroidal harmonics. It is reformulated as an optimal control problem on a fixed annular domain of external boundary $\Gamma_O$ and fictitious inner boundary $\Gamma_I$. A regularized Kohn-Vogelius cost function depending on the value of the flux on $\Gamma_I$ and measuring the discrepency between the solution to the equation satisfied by the flux obtained using Dirichlet conditions on $\Gamma_O$ and the one obtained using Neumann conditions is minimized. The method presented here has led to the development of a software, called VacTH-KV, which enables plasma boundary reconstruction in any Tokamak.