A dialysis machine for treating a body fluid, such as blood or plasma, the dialysis machine comprising: - a dialyzer (100, 100') comprising an enclosure which includes a body fluid passage zone which has a body fluid inlet and a body fluid outlet, and a dialysate passage zone which has a dialysate inlet and a dialysate outlet, and a membrane system between the body fluid passage zone and the dialysate passage zone; - a dialysate delivery system, - a connection system for connecting a dialysate supply source to the dialysate delivery system, the connection system comprising: - a first section, which comprises a conduit provided at one end with a first connection device which is connectable to a connector of a first dialysate supply source, and which is disconnectable from said connector of the first dialysate supply source, and provided at the other end with a second connection device, said conduit being a flexible conduit of constant cross section in the uncrushed state of said conduit, and - a second section, which comprises a conduit which has one end connected to the dialysate delivery system and which is provided at an opposite end with a connection device which is connected, in a disconnectable manner, to the second connection device of the first section, in order to provide a series connection of the first section and the second section, while allowing said first section to be disconnected from the second section, said conduit of the second section being a flexible conduit of constant cross section in the uncrushed state of said conduit, so that, in the state with the first section disconnected and removed from the second section, said connection device of the second section is connectable to a connector of a second dialysate supply source and disconnectable from said connector of the second dialysate supply source. - a dialyzer (100, 100') comprising an enclosure which includes a body fluid passage zone which has a body fluid inlet and a body fluid outlet, and a dialysate passage zone which has a dialysate inlet and a dialysate outlet, and a membrane system between the body fluid passage zone and the dialysate passage zone; - a dialysate delivery system, - a connection system for connecting a dialysate supply source to the dialysate delivery system, the connection system comprising: - a first section, which comprises a conduit provided at one end with a first connection device which is connectable to a connector of a first dialysate supply source, and which is disconnectable from said connector of the first dialysate supply source, and provided at the other end with a second connection device, said conduit being a flexible conduit of constant cross section in the uncrushed state of said conduit, and - a second section, which comprises a conduit which has one end connected to the dialysate delivery system and which is provided at an opposite end with a connection device which is connected, in a disconnectable manner, to the second connection device of the first section, in order to provide a series connection of the first section and the second section, while allowing said first section to be disconnected from the second section, said conduit of the second section being a flexible conduit of constant cross section in the uncrushed state of said conduit,