Electric machine having a fluid cooling device, comprising - a stator having a stator laminated core and having a stator winding, - a rotor having a rotor laminated core, said rotor being mounted so as to be rotatable about an axis relative to the stator by means of a rotor shaft and a first bearing which is arranged on a first end shield, and wherein - the fluid cooling device has a cooling tube which is fixed to a carrier element, which is fixed relative to the stator, of the electric machine and is connected to a fluid inlet and which extends axially within a central cutout of the rotor shaft and so as to form an annular space relative to the rotor shaft, wherein - the cooling tube is fluidically connected to the annular space and to a fluid outlet, and wherein - a sealing region having a sealing element is provided between the rotor shaft and the carrier element, said sealing element at least substantially preventing a transfer of fluid into a space region of the electric machine that lies outside the sealing region when a cooling fluid is flowing through from the fluid inlet to the fluid outlet, wherein - for the mounting of the rotor, a second bearing which is axially spaced apart from the first bearing is provided between the rotor shaft and a second end shield, wherein the rotor laminated core extends in a space between the first and the second bearing (13, 14), characterized in that - the rotor shaft has a shaft portion which protrudes axially beyond the second bearing and has an output element (37, 38), wherein the central cutout of the rotor shaft extends axially into the region of the output element (37, 38) and is flowed through by the cooling fluid. - a stator having a stator laminated core and having a stator winding, - a rotor having a rotor laminated core, said rotor being mounted so as to be rotatable about an axis relative to the stator by means of a rotor shaft and a first bearing which is arranged on a first end shield, and wherein - the fluid cooling device has a cooling tube which is fixed to a carrier element, which is fixed relative to the stator, of the electric machine and is connected to a fluid inlet and which extends axially within a central cutout of the rotor shaft and so as to form an annular space relative to the rotor shaft, wherein - the cooling tube is fluidically connected to the annular space and to a fluid outlet, and wherein - a sealing region having a sealing element is provided between the rotor shaft and the carrier element, said sealing element at least substantially preventing a transfer of fluid into a space region of the electric machine that lies outside the sealing region when a cooling fluid is flowing through from the fluid inlet to the fluid outlet, wherein - for the mounting of the rotor, a second bearing which is axially spaced apart from the first bearing is provided between the rotor shaft and a second end shield, wherein the rotor laminated core extends in a space between the first and the second bearing (13, 14), characterized in that - the rotor shaft has a shaft portion which protrudes axially beyond the second bearing and has an output element (37, 38), wherein the central cutout of the rotor shaft extends axially into the region of the output element (37, 38) and is flowed through by the cooling fluid.