I assume this is where it broke as the other piece obviously screws into the base stem.
[image|2710959]
I can't tell from the pic what the remedy might be, but it might be that the nut was slipped over the shaft and the end was flared. Then the silver piece was screwed into the nut. So, now the flared end has slipped out.
I would remove the lamp so that you can disconnect the wire and remove it thru the bottom. Then totally disassemble that section to see if I am right. If so cut off the flare and redo it. You will probably need a flaring tool to do it right.
Otherwise, I would remove that flare and get a piece of oak or maple and drill a hole from either side, the right size to fit snugly on top and bottom. Make sure it is long enough and thick enough to supply requires support. You may need to configure a couple of set screws.