Flue lets in water

My roofer says that the venting at the top of the furnace flue keeps out rain, but that winter snow build when no one is here gets in. The flue pipe is very rusted and corroded at the bottom in the cellar and large pools of water accumulate on the floor below the flue. He said that if the furnace is running at comfort levels the snow is melted away and disappears, but if that doesn't happen because the house is maintained just warm enough to keep the pipes from freezing, water gets in and runs down the flue pipe. Is there some kind of flue cap that protects me from this. It looks like I will have to replace the whole bottom of my flue and that doesn't solve the problem.