In this case the pith and central heartwood had rotted away, otherwise I would have drilled a central escape hole with a forstner bit. Pretty straight forward really, enlarge the hole using gouges and a mallet, followed by a spoon knife to smooth the inner surface and get the walls to about 3/8" to 1/2" thickness.
I then mark out for a rebate which will take the base and a leather fixing gasket.
Then remove about 1/4" material with a carving knife to form the rebate : this is the most tedious step.
I use 'Saral' to mark the inner profile of the bottom of the vessel onto some thin timber to make a base.
Here the base has been cut out and fitted. It is pushed up snugly against the rebate and fixed in position by a thick leather thong and some copper tacks.