A bag including two generally parallel sheets forming the main walls of the bag, complementary closure strips fixed to respective ones of the sheets, and a cursor for actuating the strips for closing and opening purposes. The bag further includes, parallel to the closure strips, between the sheets, and level with the mouth of the bag, additional structure in relief disposed on the insides of the closure strips, designed to provide sealing by forming a barrier between the sheets in the closed position of the bag, the additional structure in relief being placed facing the flanks of the cursor to be urged towards their sealing position by the cursor when the cursor is moved towards the sealing position.