The invention relates to a valve (3) that comprises a housing (20) and a thermostatic member (52) having a mobile piston (58) that carries a plug (76) for controlling the fluid flow between two accesses to the housing. In order to simplify the manufacture and the maintenance of the valve, inter alia, a hollow body (54) bearing the thermostatic member is adapted both to be rigidly connected to the fixed portion of the thermostatic member so that the piston can extend inside the inner chamber (64) of the body and so that the plug controls the fluid flow through said chamber, and to be received and sealingly blocked in an inner volume (V20) of the housing so that the fluid cannot flow between the two accesses through the chamber, while a thermostatic assembly (50), defined by the bearing body and the thermostatic member connected together, can be urged as a whole into the inner volume of the housing.