A stirring mill comprising a horizontally arranged milling vessel, which has a cylindrical inner wall, comprising a stirring shaft, which is arranged in the milling vessel and which can be driven about a common central longitudinal axis in a drive direction of rotation, comprising a milling chamber, which is limited by the inner wall and the stirring shaft, wherein, at a first end of the milling chamber, a milling material inlet discharges into said milling chamber, wherein a milling material outlet opens out from a second end of the milling vessel opposite the first end of the milling vessel, wherein the stirring shaft has a first stirring shaft section of a smaller diameter D24 and of a length L24, which is adjacent to the milling material inlet and which limits a first milling chamber region, and a second stirring shaft section of a larger diameter D25 and of a length L25, which is adjacent to the milling material outlet and which limits a second milling chamber region, wherein the following applies: D25 > D24, and wherein a transition section, which is assigned to the first stirring shaft section, is formed between the first stirring shaft section and the second stirring shaft section, wherein the second stirring shaft section has a hollow space, which is closed by a bottom with respect to the first stirring shaft section and which has a length L27 towards the first stirring shaft section, and in which a screen is arranged, which is connected to the milling material outlet and which ends at a distance upstream of the bottom, and wherein the second stirring shaft section has the hollow space comprising slits, which connect the hollow space to the second milling chamber region, wherein the length L27 of the hollow space is smaller in the direction towards the first stirring shaft section than the length L25 of the second stirring shaft section, characterized in that the first stirring shaft section has at least one bypass channel (39, 39', 39"), which penetrates the bottom of the hollow space and which connects the first milling chamber region to the hollow space and which is at least partially formed in the transition section, and that the at least one bypass channel (39, 39', 39") is inclined radially to the outside opposite to the drive direction of rotation. comprising a horizontally arranged milling vessel, which has a cylindrical inner wall, comprising a stirring shaft, which is arranged in the milling vessel and which can be driven about a common central longitudinal axis in a drive direction of rotation, comprising a milling chamber, which is limited by the inner wall and the stirring shaft, wherein, at a first end of the milling chamber, a milling material inlet discharges into said milling chamber, wherein a milling material outlet opens out from a second end of the milling vessel opposite the first end of the milling vessel, wherein the stirring shaft has a first stirring shaft section of a smaller diameter D24 and of a length L24, which is adjacent to the milling material inlet and which limits a first milling chamber region, and a second stirring shaft section of a larger diameter D25 and of a length L25, which is adjacent to the milling material outlet and which limits a second milling chamber region, wherein the following applies: D25 > D24, and wherein a transition section, which is assigned to the first stirring shaft section, is formed between the first stirring shaft section and the second stirring shaft section, wherein the second stirring shaft section has a hollow space, which is closed by a bottom with respect to the first stirring shaft section and which has a length L27 towards the first stirring shaft section, and in which a screen is arranged, which is connected to the milling material outlet and which ends at a distance upstream of the bottom, and wherein the second stirring shaft section has the hollow space comprising slits, which connect the hollow space to the second milling chamber region, wherein the length L27 of the hollow space is smaller in the direction towards the first stirring shaft section than the length L25 of the second stirring shaft section, characterized in that the first stirring shaft section has at least one bypass channel (39, 39', 39"), which penetrates the bottom of the hollow space and which connects the first milling chamber region to the hollow space and which is at least partially formed in the transition section, and that the at least one bypass channel (39, 39', 39") is inclined radially to the outside opposite to the drive direction of rotation.