The evolution of collapsing primordial clouds and the formation of Population III (Pop III) protostars are investigated with three-dimensional ideal MHD simulations. We follow the collapse of parsec-sized primordial clouds down to the formation of protostars on sub-AU scales. Pop III protostar formation is characterized by the ratio of rotational to magnetic energy of the natal cloud. When the rotational energy is larger than the magnetic energy, fragmentation occurs prior to the formation of the protostar, and a binary or multiple system appears. When the magnetic energy is greater than the rotational energy, a strong (> 100 km s-1) jet is driven by the circumstellar disk around the protostar, which does not fragment. Thus, even in the early universe, magnetic fields play an important role in the star formation process.