A band saw machine comprising a band saw and a pair of saw wheels (5a, 5b), wherein the band saw is wound around the pair of saw wheels(5a, 5b) disposed spaced apart, the band saw including a teeth edge portion formed of a plurality of saw teeth laying side-by-side in a circumferential direction, wherein each of the saw wheels (5a, 5b) includes, at an outer periphery, a rim portion around which the band saw is wound, and rotates around a rotary shaft extending in a thickness direction, wherein the rim portion includes: a supporting surface configured to support the band saw; an edge provided at an end portion on at least one side of the rim portion in an axial direction of the rotary shaft, and configured to be contacted by the band saw in a state where the teeth edge portion is projected from the end portion; a non-contact area (5332b, 5333, 5334, 5362b, 5361, 5331) disposed between the supporting surface and the edge, and configured not to contact the band saw; a contact surface (5332a, 535, 5362a) disposed between the edge and the non-contact area (5332b, 5333, 5334, 5362b, 5361, 5331) and configured to be contacted by the band saw, characterized in that the rim portion includes either a first groove or, alternatively, a notch, wherein one of the following options applies: • the rim portion includes the first groove provided at an end portion on the one side and extending in a circumferential direction, the non-contact area (5332b, 5331)and the contact surface are disposed at predetermined positions of the first groove, and the first groove has a width which is 14 times or more and 23 times or less of a thickness of the band saw, and a depth which is 3/1000 or more and 20/1000 or less of the thickness of the band saw, or • the rim portion includes the notch formed by cutting out an end portion on the one side in a circumferential direction, the non-contact area (5362b, 5361) and the contact surface are provided at predetermined positions of the notch, and the notch has a length which is 14 times or more and 23 times or less of a thickness of the band saw in the axial direction, and a depth which is 3/1000 or more and 20/1000 or less of the thickness of the band saw. a supporting surface configured to support the band saw; an edge provided at an end portion on at least one side of the rim portion in an axial direction of the rotary shaft, and configured to be contacted by the band saw in a state where the teeth edge portion is projected from the end portion; a non-contact area (5332b, 5333, 5334, 5362b, 5361, 5331) disposed between the supporting surface and the edge, and configured not to contact the band saw; a contact surface (5332a, 535, 5362a) disposed between the edge and the non-contact area (5332b, 5333, 5334, 5362b, 5361, 5331) and configured to be contacted by the band saw, characterized in that the rim portion includes either a first groove or, alternatively, a notch, wherein one of the following options applies: • the rim portion includes the first groove provided at an end portion on the one side and extending in a circumferential direction, • the rim portion includes the first groove provided at an end portion on the one side and extending in a circumferential direction, the non-contact area (5332b, 5331)and the contact surface are disposed at predetermined positions of the first groove, and the first groove has a width which is 14 times or more and 23 times or less of a thickness of the band saw, and a depth which is 3/1000 or more and 20/1000 or less of the thickness of the band saw, or • the rim portion includes the notch formed by cutting out an end portion on the one side in a circumferential direction, • the rim portion includes the notch formed by cutting out an end portion on the one side in a circumferential direction, the non-contact area (5362b, 5361) and the contact surface are provided at predetermined positions of the notch, and the notch has a length which is 14 times or more and 23 times or less of a thickness of the band saw in the axial direction, and a depth which is 3/1000 or more and 20/1000 or less of the thickness of the band saw.