The distribution box has a shape slightly resembling that of a frustum of pyramid, and its removable diaphragms, with outer rings (9, 10) for the insertion of pipes, are made in recessed parts (7, 8) of the side walls of the distribution box and do not protrude from the said recesses, which are preferably open at the bottom and have a width that increases downwards. The inside of the distribution box is thus characterized by presenting parts that are in relief corresponding to the said external recesses (7, 8), on which are inserted the bottom open and free parts of the same external recesses (7, 8) when a number of distribution boxes of the type referred to are piled up so that they fit vertically into one another. To ensure proper and stable stacking of the distribution boxes, the latter are provided with vertical, external stiffening ribs (14), appropriately distributed and of equal height, with which one distribution box rests on the top rim (101) of the distribution box beneath it, when a number of distribution boxes are piled up so that they fit vertically into one another.