Trim any damaged material from the bottom with clean cuts from a very sharp knife, and place it on its side in warm shade with plenty of air circulation for about a week so the cut can "callus" or heal. Fill a small container with a mix of 75% perlite and 25% potting soil. Dampen the mix slightly so that it doesn't separate as you mix it. It's better if the container is taller than it is wide. Place the cactus just deep enough so that it is stable, and put the container in a sunny (but not hot) windowsill. The soil will probably have enough moisture to last until the plant produces roots, so wait until you see new growth before watering. When it roots, you'll see bright new growth in the center. You can set it outside on sunny days after that, but I'd wait until spring before feeding it or leaving it outside.
Most people can't help themselves and will be tempted to occasionally lift the plant and look for roots. If you put about an inch of perlite on top of the soil mix and set the cutting into that, you will be able to gently lift the plant every couple of weeks to check. The perlite will make it easy to remove and replace the cutting.
Edit: It appears that a little cross-posting happened...
