Cut the original key to round the very end just a bit.
Place the key in the lock and turn 90 degrees. The cut key moves the entire shaft inside the lock instead of just the tumblers.
Push the retaining wire into the small slot next to the keyhole until it is fully seated. This action opens the padlock.
Push the shackle back into the hole as if you are locking it again and at the same time pull on the key. Wriggle the key until the entire cam inside the lock comes out.
Insert the new shaft into the padlock, lining up the slot on the shaft with the retaining wire still in the lock. Push the new shaft in until it is flush with the bottom of the lock.
Remove the retaining wire, making sure to avoid pushing the shackle into the hole and locking the padlock (See Reference 2).
Insert the new key into the lock. The key will only go partway in at this point. Close the lock by putting the shackle into the hole and turn the key until the keyhole is back to its original position, parallel to the shackle.