Hello Alex, If the bad area at the base of the plant is the only questionable section I would begin cutting an inch or so above the rotting trunk until there was no rot showing on the cross-section of my cut. Sharp shears cleaned after each cut with peroxide or alcohol will help prevent the infection from returning to the salvaged cutting. I would probably wait until spring to root the cutting to up the odds of survival of the plant. Remove all full grown leaves by cutting them off at about 3/4" from the trunk.

Great, thank you! How should I preserve the cutting in the meantime?

Put rooting hormone on the cut area; it contains fungicide, and store until spring in a dry area protected from freezing temperatures.