A contact support suitable for mounting on a cycle brake control cable, the support comprising: a first portion which is fixed relative to the cycle frame, said fixed portion having a first contact surface; a second portion which is movable relative to the cycle frame, and suitable for being secured to the cable, said moving portion presenting a second contact surface; the support comprising connection means between the fixed portion and the moving portion, said connection means being suitable for being deformed elastically between: an extreme relaxed state for the support in which the contact surfaces are spaced apart from each other; and a tensioned state for the support in which the contact surfaces press against each other, the connection means then being elastically deformed.