Answer: First of all, the sprout you see may not be the same variety as the tree you had. Fruit trees are often grafted -- that is, they are composed of two different varieties. The upper part, called the scion -- the part that produces the fruit -- is grafted onto a rootstock. Sometimes the rootstock of a tree will sprout even when the upper part of the tree is still alive. If what you have is a root sucker from a rootstock, then it won't be the same variety as the original tree. However, if your tree was grown from seed, rather than grafted, then it will indeed be the same variety.
Either way, the sprout may grow into a healthy tree, or it may succumb to the same disease that rotted the original tree. You could wait it out and see what happens, but if you want to ensure you'll get a tree that produces quality fruit, I'd suggest starting over with a new sapling tree.
If you decide to nurture the sprout, clean up under the tree to remove brush or tall grass and place a thick mulch on the soil surface beneath the branch spread to reduce weed competition. When the sapling begins to regrow in spring, make sure and thin out the multitude of side shoots, leaving only a few (which are oriented in a desirable direction) to remake the new branching system of the tree. Because of the extensive existing roots system, it's likely you'll get a strong flush of growth in the spring, so you probably don't need to fertilize.
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