This paper demonstrates that “completely-jigless” assembly of a model product that requires fitting accuracy at the level of industrial products is possible by using a universal hand with four parallel stick fingers mounted on a conventional position-control-based industrial robot. Assuming that each part is taken out of the parts bin and temporarily placed on the work table, the accuracy required for precise fitting cannot be achieved with a vision sensor alone. Introducing an appropriate grasping strategy, the initial position error of the part is absorbed by self-alignment in the process of grasping. Once the alignment is completed, the pose of the grasped part is fixed and jigless assembly is possible with a conventional industrial robot, which has high repeatability. In this paper, we use a gear unit as an example of an industrial product and present some grasping strategies with the universal hand. We also propose some subsequent assembly strategies for shafts and gears. Using those grasping and assembly strategies, it is shown that jigless assembly of the gear unit was successfully completed in the experiment. Although the target product in this paper is specific, the assembly elements in this product, such as shaft screwing, bearing insertion, and gear meshing, are also included in many other products. Therefore, the methods shown in this paper can be applied to other products.