The rot externds further than is obvious. The living fungi are inside the fairly solid wood. Look up repairing wood rot with Borates. I bought some Bor-8 Rods. Small rods of solid borate. Drill a 1/4" hole, insert the rod and then make your repair. The borate should infuse the wood from the inside and arrest the decay. Lots of borate based kits out there.
You can also try making your own out of stuff like 20 Mule Team Borax and Antifreeze. The problem being you can only tell if it works years later.