An electric power-steering apparatus, comprising: a housing that is supported by and fastened to a vehicle body or a portion that is fastened to the vehicle body; a rotating shaft that comprises a front-end section that is supported by the housing so as to be able to rotate freely, and that applies a steering angle to steered wheels, and a rear-end section that is connected to a steering wheel; a worm wheel that is concentrically supported by and fastened to part of the rotating shaft on the inside of the housing; a worm that comprises a worm shaft that has a base end section and worm teeth provided around the middle section in the axial direction thereof, and, with the worm teeth engaged with the worm wheel, is supported by the housing so as to be able to rotate freely; and an electric motor that rotates and drives the worm, and comprises: a motor case that is supported by and fastened to the housing, an output shaft that has a tip end section that is connected to the base end section of the worm shaft so that torque can be transmitted, and a pair of single-row ball bearings (14a, 14b) that support the output shaft on the inside of the motor case so as to be able to rotate freely; each of the pair of single-row ball bearings (14a, 14b) comprising an outer ring (24a, 24b) that is supported on the inside of the motor case in a state where displacement in a direction axially away from the worm is restricted, and an inner ring (28a, 28b) that fits around a part of the output shaft in a state where displacement in a direction axially toward the worm is restricted; the base end section of the worm shaft comprising a base end surface, and an engagement hole that has a non-circular cross sectional shape, is open on the base end surface, and has a back end surfaces; the tip end section of the output shaft comprising an engagement rod section that has a non-circular cross-sectional shape, engages with the engagement hole and has a tip end surface; and an elastic member (31, 32) that has an elastic force in the axial direction being held in an elastically compressed state in the axial direction between the back end surface of the engagement hole and the tip end surface of the engagement rod section, characterized in that by applying an elastic force to the output shaft in a direction axially away from the worm, the value of which is greater than the value of a friction resistance to displacement in the axial direction of an engagement section between the engagement hole and the engagement rod section in a state where the friction resistance becomes a maximum by the torque that is transmitted by an engagement section between the worm wheel and the worm becoming maximum, the reaction force in the radial direction that is applied to the worm from the engagement section between the worm wheel and the worm becoming a maximum, and the radial load that is applied to the engagement section between the engagement hole and the engagement rod section, applies a preload at the same time, together with a tandem arrangement contact angle, to the pair of single-row ball bearings (14a, 14b), and at least one of the outer rings (24a, 24b) comes in contact with a stepped surface (27, 29) on the inner circumferential side that is formed in part of the motor case by way of a ring-shaped plate spring, the thickness in the axial direction thereof being able to expand and contract, or by way of a shim plate that is selected from among many kinds of shim plates having different thickness dimensions. a housing that is supported by and fastened to a vehicle body or a portion that is fastened to the vehicle body; a rotating shaft that comprises a front-end section that is supported by the housing so as to be able to rotate freely, and that applies a steering angle to steered wheels, and a rear-end section that is connected to a steering wheel; a worm wheel that is concentrically supported by and fastened to part of the rotating shaft on the inside of the housing; a worm that comprises a worm shaft that has a base end section and worm teeth provided around the middle section in the axial direction thereof, and, with the worm teeth engaged with the worm wheel, is supported by the housing so as to be able to rotate freely; and an electric motor that rotates and drives the worm, and comprises: a motor case that is supported by and fastened to the housing, an output shaft that has a tip end section that is connected to the base end section of the worm shaft so that torque can be transmitted, and a pair of single-row ball bearings (14a, 14b) that support the output shaft on the inside of the motor case so as to be able to rotate freely;