You can certainly behead the plant if the top part is still healthy. Use a sharp knife and be sure to leave a reasonable margin above the rotten part when you cut. If the terminal rosette is healthy at the stem level then it should be able to make it through.
Allow the cutting to heal for a few days out of the sun, then pot it up with a half an inch of stem below the rosette ideally poking into the soil. Use a nice fast draining mix with maybe half rock (pumice, perlite or equivalent). Don't water for a week or more. Then resume normal care (watering when the soil is dry or almost dry) and provide strong light. You will need to be patient. When the cutting grows roots there will be a visible change on top as it rejuvenates and begins to sprout new leaves. At that point you know you are out of the woods. I would guess you have less than even odds for recovery but I love to prove these sorts of predictions wrong.
