  This paper is concerned with an inverse obstacle problem for the Laplace's
equation. The aim is to recover the constant conductivity coefficient in the
equation and the boundary of a Dirichlet polygonal obstacle from a single pair
of Cauchy data. Uniqueness results are established under some a priori
assumptions on the input boundary value data. A domain-defined sampling method,
based on the factorization method originating from inverse acoustic scattering,
has been proposed to recover both the constant conductivity coefficient and the
polygonal obstacle. A hybrid strategy, which combines the sampling method and
iterative scheme, is employed {\color{hgh}to reconstruct} the location and
shape of the obstacle. Numerical examples indicate that our method is
efficient.
