A method of connecting chips to chip carriers, ceramic packages, etc. (package substrates) forms smaller than usual first solder balls and polymer pillars on the surface of a semiconductor chip and applies adhesive to the distal ends of the polymer pillars. The method also forms second solder balls, which are similar in size to the first solder balls, on the corresponding surface of the package substrate to which the chip will be attached. Then, the method positions the surface of the semiconductor chip next to the corresponding surface of the package substrate. The adhesive bonds the distal ends of the polymer pillars to the corresponding surface of the package substrate. The method heats the first solder balls and the second solder balls to join the first solder balls and the second solder balls into solder pillars.