The device includes two sub-assemblies (1) for mounting the rail (2) for supporting the curtain on the leaves (3) of the window, each sub-assembly having a plate (5) and an offset arm (6). The plate is formed so as to be able to be fixed to the upper edge of a leaf (3), in the vicinity of the pivot axis thereof, and includes mounting elements (14) for pivotably mounting the offset arm thereon, along an axis extending, after mounting on the leaf (3), parallel to the pivot axis of the leaf. The offset arm (6) is formed to offset the rail (2) relative to the wall; it is pivotably mounted on the plate (5) by the mounting elements and is pivotably connected to the rail (2) at a point thereof which is longitudinally fixed, also along an axis parallel to the pivot axis of the leaf (3).