The invention relates to a method and an apparatus for filling a container which is open at the top with a pourable product. A filling head is placed on the container. The container is received in a gas-tight sleeve which, at least in a filling position, adjoins the filling head and/or the container in a sealing manner to provide a gap between the sleeve and a container outer wall. For a product discharge, a reduced pressure up to a first pressure level is produced in the container and in the gap. After the product discharge is finished, a gas supply to the container and to the gap is carried out. Prior to the gas supply, a reduced pressure up to a second pressure level is produced, for a degassing of the container. The second pressure level is below the first pressure level.