Find and obtain a long piece of rope. The rope should be long enough that no one will notice when you cut a few inches from it. Put one end of the rope in the crease where your left thumb meets your left palm, and take the other end in between your left index finger and your left middle finger. There should be an inch or so of each end of the rope sticking up above your hand, so the audience can see the ends.
Take the bottom of the loop of the rope in your right hand, and move your right hand to your right side, so the rope loop is parallel to the ground. From below, put your right thumb and index finger into the loop, so they are pointing upward.
Move your right hand towards your left, and in a fluid motion, use your right index finger and thumb to pinch the part of the rope that is resting in your palm (the string dangling from your left thumb, not from in between your left index and middle finger).
Pull your right hand up, so the back of the loop slips off your hand and hides behind your palm. You should now be holding part of the string that will appear to be the "center" of the rope. Form a loop as you pull it upward. Pull it up so the audience can see the "loop." They will think this is the real center, although the real center is hidden in your palm.
Hold the loop you have just created, using your thumb to press it against your fingers. Using scissors cut the middle of the loop. You should now have four ends of rope sticking up above your fingers.
Hold the two ends closest to you with your thumb, and let the two farthest away from you drop. This should reveal that the rope is indeed in two relatively equal pieces (although in reality, it is not). Still hiding the real "intersection" of the two ropes behind your palm, tie the small piece of rope onto the longer one. This will appear as if you have tied the two "equal" pieces together, and you can now let the audience see the whole rope.
Say a few magic words, gather the rope up in your palms, and slide the small piece from the larger one, merely by sliding the knot. Hide the small piece in your hands as you show off the "newly restored" piece of longer rope.