Given a closed positive current T on a bounded Runge open subset Ω of Cⁿ, we study sufficient conditions for the existence of a global extension of T to Cⁿ. When T has a sufficiently low density, we show that the extension is possible and that there is no propagation of singularities, i.e. T may be extended by a closed positive C^∞-form outside $\bar \Omega $ . Conversely, using recent results of H. Skoda and H. El Mir, we give examples of non extendable currents showing that the above sufficient conditions are optimal in bidegree (1, 1).