Needle just happens to be the right size to fit the plastic tubing I already had on hand.
Drilled a hole through the dowel, into the socket. Put the needle in, and siphoned hot water from the teakettle into the joint.
There's the teakettle, on an upside down saucepan, to give it a little better height for the siphon to run, and a closeup of the needle into the dowel.
The neck is held in a padded vise, sittnig in the sink. And it's slanted to the water will drip AWAY from the varnish on the neck.