There's a margin where rot might not begin, but it will continue. If you're talking about housebuilding carpentry attached to a damp masonry wall, then this could be a problem for you. It also depends on the type of rot you have, and a little on the timber species.
If you're talking about storage damage on timber, then just saw it off and don't worry. You shouldn't be storing timber at anything like the humidity needed for rot to be an issue.
Largely though, any timber, including treated, will rot if the moisture conditions are right. You _must_ remove the moisture to stop this. Chemical rot treatment achieves little on rotten wood, it's mainly a way of putting a guard zone around on the good stuff. Fungal rots will propagate far better on the surface of masonry than on timber, even though these "threads" might not themselves be damaging.
It will likely continue. The moist environment is not what is rotting the wood. It is what breeds in the moist environment that eats the wood. Remove the moisture and the rot may slow down but likely it will continue unless you kill the organisms that are eating away at the wood.
My guess would be that it would continue to rot...
Back when I was selling homes, if the pest control inspector found wood damaged by water, they would stop the source of the water (leaky eaves, plumbing leak, etc.), and then treat whatever wood that they didn't remove with something called "copper green"..
Lack of moisture should kill them, no? I've cut the spalted section off a maple or birch log and used the rest without problems. You just need to make sure you got it far enough in, otherwise the stuff you can't see will have weakened the grain lines. I remember turning a leg out of birch that was a bit stained by the beginnings of rot. After I was done, I twisted it, and it came apart along the grain.
Copper sulfate: Old, reasonably effective anti-fungal treatment. Works on live plants, too.
Responding to an earlier post, just drying won't necessarily kill fungi. They're tenacious beasts. When conditions deteriorate (for them), they just spore up and wait it out.
Just to make things grimmer, they're eukaryotes; almost anything that'll kill them will be toxic to you, too. (Yeah, Micatin, but you don't want systemic exposure to it, either, sez my pharmacist.)
Yeah, but wood with a normal moisture content doesn't rot, so it stands to reason that whatever critters cause the wood to fall apart need more moisture than that.
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