A sealing box for a chamber for continuously treating a thin strip product, in particular for a furnace for continuously carbonizing a fiber substrate. The sealing box comprises: a longitudinal passage (12) opening out from the box via a first end (12b) for connection to an inlet or an outlet of a treatment chamber (2) and via a second end (12a), opposite from the first; a support surface (14a) inside the passage, on which a strip product (T) can travel between the ends of the box; and static sealing means (30) acting by making contact with the strip product travelling along the passage on the support surface. The static sealing means comprise at least one inflatable gasket (32) placed across the passage (12) above the support surface (14a), and dynamic sealing means (40) are also provided in the passage between the second end (12a) of the box and the static sealing means, the dynamic sealing means comprising means (52, 56) for injecting gas into at least one chamber (42, 46) formed in the passage. The sealing box is suitable in particular for a furnace that produces carbon fiber cloth by continuously carbonizing a cloth made of a carbon precursor.
