An exhaust turbocharger has a housing, in which a shaft is mounted by bearings, which carries a turbine wheel, on the one hand, and a compressor wheel, on the other hand. An electric machine has a rotor and a stator. The rotor is secured on the shaft for conjoint rotation therewith and the stator surrounds the rotor radially on the outside. A sleeve, which supports the rotor radially and axially, is arranged between the stator and the rotor. The sleeve has at least one fluid duct, by way of which a cooling fluid can be guided toward the bearings of the shaft.