Thanks for the question. I need a clarification.  Are these sprouts coming from a looose piece of wood or from the stump (still in the ground) of the Ginkgo? 
Looking forward to your response. 

It is a loose piece of wood. I took the picture where it is sitting on a cement step. 

Thanks for the question and your subsequent clarification.

I suspect that what is going on is this new sprout is getting its nourishment from stored energy in the piece of wood. In other words, I doubt that this sprout has any root system of its own. It also means that if it remains attached to the piece of wood, it will soon exhaust its source of energy. The only suggestion that I can make, albeit one in which I have relatively little confidence, is to cut the sprout off next to the bark of the wood piece and place the sprout in a glass of water in hopes that roots would emerge from the spout. Should that occur, the sprout with its roots could be placed in a pot containing potting soil and then see what happens.

Thanks for the question. I wish that I could be more positive in my response. Good luck.