Given a closed positive current$T$on a bounded Runge open subset Ω of$C$^{$n$}, we study sufficient conditions for the existence of a global extension of$T$to$C$^{$n$}. 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).