Summary: We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset \(\Gamma _{+}\) and all the corresponding Neumann data measured on an open subset \(\Gamma _{-}\). We prove the global uniqueness under some additional geometric condition, in the case where \(\overline{\Gamma _+ \cap \Gamma _-} = \emptyset \), and we prove also the uniqueness for a similar inverse problem for the stationary Schrödinger equation. The key of the proof is the construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.