An electric power steering apparatus for an automobile includes a reduction mechanism having a worm shaft and a worm wheel, the worm shaft having a worm formed on an outer peripheral surface of the worm shaft, the worm shaft being contained in a housing, the worm wheel being formed on a side of the steering shaft, the reduction mechanism being driven by the motor; a support member for enclosing an end-side outer peripheral surface of the worm shaft; and an elastic member for applying elastic force to the support member in a direction slanted away from an axial direction of the worm shaft. The gap between the worm wheel and the worm shaft is compensated for and backlash is avoided. Since a single elastic member is used, the number of components is reduced, the manufacturing process is simplified, and the manufacturing cost is reduced.