A superconducting planar undulator with 9.5 mm pole gap, 16 mm period and required peak field 0.64 T is at present under development at the Advanced Photon Source. Magnet structure precision is a key component of the project. The current design of the magnet structure is based on the assembled jaws with individual poles, while previous designs utilized solid cores with machined coil grooves. Each jaw has a core with pole grooves and separate precise poles inserted in these grooves and fixed with small screws. This approach allows achieving a uniformity of the poles thickness to within 15 µm, improving the surface finish, as well as using magnetic poles with non-magnetic cores. Several 42-pole jaws have already been produced, wound and tested. Magnetic measurements of the first structure have confirmed the mechanical precision of the assemblies. Details of the magnet structure design and jaw assembly, and changes made from the first prototype system to the production unit, are presented here.