Doc's suggestion is the best and permanent solution.
But if you are game for a little mad-scientist fun, and really just want to copy data, then you could set up the 2 boxes side by side, use the power connector from the old box to power the old hard drive, then connect the SATA cable to the motherboard in the new PC (both boxes OFF!) [you may need to stand the old hard drive on the table to "bridge" the 2 PC's]
Power up the old PC (it will sit on a black screen and cry about "no hard drive!" - don't worry, it won't hurt anything)
Then power up the new PC. It should recognize the second drive and you are good to go (copying data)!
If you are uncomfortable with any of this, or really want a permanent 2-drive solution, then just order the "Molex to Sata Power Dongle" and do it the "correct" way!