Assembly comprising a cap and a container that is able to contain a fluid, said container comprising a neck that has an external thread and an annular end rim defining a pouring opening through which the fluid can be poured out of the container said cap being arranged so as to be able to move on the neck between a closing position, in which fluid is prevented from passing through the pouring opening, and at least one pouring position, in which fluid can be poured out of the container through the pouring opening, said cap comprising: - an end wall having an external face and an internal face and having a through-opening, - a peripheral skirt extending from and away from the internal face of the end wall in a first axial direction (I-I), and comprising an internal thread that is able to cooperate, by screwing, with the external thread of the neck of the container, - first sealing means essentially in the general shape of a disc having a first face, termed upper face, a second face, termed lower face, and having a peripheral edge, these means being intended to seal closed the pouring opening by pressing against the annular end rim when the cap is in the closing position, said first sealing means being applied against the internal face of the end wall, at least one peripheral passage being permanently created between the peripheral edge of the first sealing means and the peripheral skirt, in which, in the pouring position, first sealing means allow fluid to pass in a conveying path comprising a space between the annular end rim and the second face of the first sealing means, said at least one peripheral passage and one space between the first face of the first sealing means and the internal face of the end wall, said conveying path extending as far as the through-opening, characterized in that the cap comprises second sealing means located away from the internal face of the end wall and beyond said at least one peripheral passage in the first axial direction (I-I), which means come, for any position of the cap from said at least one pouring position into a closing position, into sealing contact around a distal cylindrical portion from the neck, which distal cylindrical portion extending from and away from the pouring opening and being located between the pouring opening and the external thread of the neck. - an end wall having an external face and an internal face and having a through-opening, - a peripheral skirt extending from and away from the internal face of the end wall in a first axial direction (I-I), and comprising an internal thread that is able to cooperate, by screwing, with the external thread of the neck of the container, - first sealing means essentially in the general shape of a disc having a first face, termed upper face, a second face, termed lower face, and having a peripheral edge, these means being intended to seal closed the pouring opening by pressing against the annular end rim when the cap is in the closing position, said first sealing means being applied against the internal face of the end wall, at least one peripheral passage being permanently created between the peripheral edge of the first sealing means and the peripheral skirt, in which, in the pouring position, first sealing means allow fluid to pass in a conveying path comprising a space between the annular end rim and the second face of the first sealing means, said at least one peripheral passage and one space between the first face of the first sealing means and the internal face of the end wall, said conveying path extending as far as the through-opening, characterized in that the cap comprises second sealing means located away from the internal face of the end wall and beyond said at least one peripheral passage in the first axial direction (I-I), which means come, for any position of the cap from said at least one pouring position into a closing position, into sealing contact around a distal cylindrical portion from the neck, which distal cylindrical portion extending from and away from the pouring opening and being located between the pouring opening and the external thread of the neck.