**UPDATE** I have since made a video tutorial on how to do this and it might be more helpful than the following procedure below. To watch that tutorial click here: https://www.youtube.com/watch?v=BLD0yp9Rpbk
1. Disassemble Your Rubik's Cube. (here's a helpful video) http://www.youtube.com/watch?v=cM1Jbzy5Pnk&list=SPFC43E5797C311AC3
2. Find a glass jar that you can fit the core of the cube into, but can't fit the whole cube into it when it the cube was completely assembled. (make sure the inside of the jar (not the opening but the hollow space inside) isn't that small so that you can do the next step easily, and so that it will have enough room to shake around and not just be jammed in there when it's assembled.)
3. Put the core of the Rubik's Cube into the jar and assemble the cube as show in the YouTube video above, inside the jar. When the first 2 layers of the cube are done being assembled, I recommend placing the jar on a table with the opening facing up, and now drop the next piece in and use something like a chopstick (or chopsticks or any stick type thing) to correctly orient the piece on top of the cube and then pop it in with your finger (or the chopstick if your finger isn't long enough).
Doing this whole process is harder if your cube is tight. If your cube is loose it will work a bit easier. To loosen your cube if it's tight, lubricate your cube with Vaseline (petroleum jelly), and then after a year or two or three of twisting, turning, and solving, the Vaseline will have dissolved a bit of plastic off the inside of your cube making it a bunch looser. That's why it wasn't extremely hard when I did it, but it was still pretty tricky.
Watch a video of this here http://www.youtube.com/watch?v=kvAi9Uvm3xY