John,
I bought a house with a tree that sounds much like yours once. The previous owner had filled it with concrete, and I didnt' notice until too late. During a storm one night, the tree split right down the middle top to bottom. What I didn't know then is that concrete doesn't give when the tree moves and sways. Instead, it sort of grinds the tree from the inside out like a giant rasp.
You should probably call in a professional arborist, as the heart rot is actually caused by fungi. They might be able to kill that off and then recommend an appropriate course of action to take to save the rest of the tree (if possible).