Disclosed is a transmission housing (1) comprising a conical friction clutch (2), a shaft (3) carrying said clutch (2), and a control device (4) for the friction clutch (2); the clutch (2) includes at least one assembly, each of which comprises a male cone (71; 72) that has a conical frictional surface and is mounted on the shaft (3) carrying the clutch (2) for conjoint rotation therewith, and a guiding female conical bearing surface (51; 52); the male cone (71; 72) of the assembly is hollow in the center so that it can be placed on the shaft (3), and includes at least one recess (11) along the peripheral edge delimiting the hollow center; the female conical bearing surface (51; 52) and the associated male cone (71; 72) of the assembly can move from a disengaged position under the effect of the control device (4). In order to securely connect the male cone (71) or at least one of the assemblies to the shaft (3) for conjoint rotation therewith, the shaft (3) has one or more radial protrusions (81) that are made as a single part along with the shaft (3) and can each engage a recess (11) in the male cone (71).Boîtier (1) de transmission comprenant un embrayage (2) à friction à cônes, un arbre (3) porteur dudit embrayage (2) et un dispositif (4) de commande dudit embrayage (2), ledit embrayage (2) comprenant au moins un ensemble, le ou chaque ensemble comportant un cône (71; 72) mâle, à surface de friction conique, monté solidaire en rotation de l'arbre (3) porteur de l'embrayage (2), et une portée (51; 52) conique femelle menante, le cône (71; 72) mâle dudit ensemble, évidé centralement pour pouvoir être positionné sur l'arbre (3), présentant, au niveau du bord périphérique de délimitation de l'évidement, au moins une encoche (11), la portée (51; 52) conique femelle et le cône (71;72) mâle associé dudit ensemble étant aptes, sous l'action du dispositif (4) de commande, à passer d'une position débrayée. L'arbre (3) comprend, pour la solidarisation en rotation du cône (7