You need to cut off all wood with discoloration, until you have only white wood. Wipe your cutting tool with alcohol or peroxide between cuts. Powder the end with rooting hormone with fungicide. Then let the cutting dry in a cool, dark place for a week or so and try to root it again.
Mike

Sounds like maybe you were watering it too much. Once you get it going again, let it dry out totally between each watering. Some say water once a week. I think it depends on where you live, temp, rain, and things like that. Just don't over water by keeping the soil wet all the time. Let it dry out.
Lauri

I agree with what Mike said. I've cut rotted stems until I only had a couple inches of healthy tissue and have some nice plants from such little sticks. I would water once when I first potted it then wouldn't water again at all until you are sure you have root development. You can mist the developing leaves but that's all I'd do. Good luck.
Karyn