Answer: Weeping cherries are grafted trees, where a top weeping piece is literally attached to a normal upward growing trunk. Based on your description, I suspect that the rootstock portion of the tree has sent out a branch from below the graft point. This would be the piece growing straight up. This wayward piece should be removed as should any growth that ever appears below the graft -- as soon as you notice it. If caught early, the tiny sprouts can even be rubbed off with your fingers. It is important to remove these growths so that the tree does not divert any energy away from the grafted top.
Another possibility is that it is a "water sprout" growing from one of the weeping branches. Again, it is best to remove these as soon as possible so they do not detract from the overall plant.
If the shoot is still fairly small in diameter, remove it now. Otherwise, wait until late winter to cut it off. Cut it off cleanly with a sharp pruning saw or possibly a bow saw, and cut it very close to the base. Do not seal the cut with pruning paint, it is no longer recommended.
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