You'll need to cut back to sound wood, I've done this with hard-wood shutters - chisel out all the rot, then use cardboard and masking tape to seal off the void, then pour in resin mixed with fine sawdust. Once it sets it's extremely tough and you can re-drill the hole for the tap.
Airheads idea of using a larger hole-saw to take out the section is a good one, if you can centre it correctly. I'd be tempted to run a router round the top of the fresh hole to provide a stepped hole for the resin to fill in order to give it even less desire to come out.
You'll need to cut back to sound wood, I've done this with hard-wood shutters - chisel out all the rot, then use cardboard and masking tape to seal off the void, then pour in resin mixed with fine sawdust. Once it sets it's extremely tough and you can re-drill the hole for the tap.
Airheads idea of using a larger hole-saw to take out the section is a good one, if you can centre it correctly. I'd be tempted to run a router round the top of the fresh hole to provide a stepped hole for the resin to fill in order to give it even less desire to come out.