Ewwww .... Gotta kill that sucker. Eliminating the moisture stops the growth of fungus but the spores will remain viable so you have to kill them too. Bleach will only kill the surface stuff but not the internal spores. Once the moisture is gone, the spores go dormant ... waiting to return!
Happily, Boron ( disodium octaborate tetrahydrate) kills existing fungus and eliminate future infestations. This is NOT the same thing as Boric Acid. I believe 20 Mule Team Borax is the preferred stuff. At least one cup per gallon of water creates a solution which should kill the spores.
Boron is "water diffusible" so it will actually use the elevated moisture to pull the boron deep into the wood. Borates defuse through wood drawing into the highest points of moisture leaving behind the residue, which fills the cellular structure of the wood with the chemical, thus preventing decay fungi and infestations of other wood destroying pests.