A bearing to be used for a pulley in an auxiliary device for the engine of an automobile includes an outer ring 3, an inner ring 5, double rows of balls 7 and 8, retainers 9 and 10, and seals 11 and 12 which are disposed on the rear surface sides of the retainers 9 and 10 so as to be spaced therefrom thereby to form spaces for accommodating grease, respectively. The bearing for a pulley used in an auxiliary device for the engine of an automobile is arranged in a manner that the outer diameter of the bearing is 65 mm or less, the width of the bearing is 45% or less of the inner diameter thereof, the diameter of each of the balls 7 and 8 is 4 mm or less, and a distance L between the rear surfaces 19 of the retainers 9 and 10 and the inner surfaces 20 of the seals 11 and 20 are set to be 13% or more of the diameter Da of the ball.