A process for treating a gas mixture containing at least an organic compound gas or vapor and a second gas, such as natural gas, refinery off-gas or air. The process uses two sequential membrane separation steps, one using membrane selective for the organic compound over the second gas, the other selective for the second gas over the organic vapor. The second-gas-selective membranes use a selective layer made from a polymer having repeating units of a fluorinated polymer, and demonstrate good resistance to plasticization by the organic components in the gas mixture under treatment, and good recovery after exposure to liquid aromatic hydrocarbons. The membrane steps can be combined in either order.