A spindle, or an arbor? It sounds like you want to turn down an arbor that originally FIT a morse taper spindle.
It probably can be cut with carbide, but it will be hard to do so. You probably would do well to anneal it. That can often be done by heating it with a propane torch. The smooth ground surface will turn "straw" colored at a certain point. Heat it perhaps a little past that point and then let it cool slowly. This should reduce the hardness enough to make turning it easier with carbide. You probably are not going to be able to turn this on a jeweler's lathe, or at least it would take a long time to do so.