Clean out all that dead brown tissue from the pot. You should be able just to pull it out gently by hand.
Stop watering the plant, and allow the soil to dry. When the soil is dry to a depth of about an inch -- dig into it with your finger to make sure -- then give it a thorough watering, until water runs out the drainage holes at the bottom of the pot. The plant will look kind of scruffy for awhile, but aloes grow fairly rapidly. After awhile, it should send up pups, filling any empty spaces in the pot.
For the part that has come off, use a clean, sharp knife to cut off the bottom of the stem, above any black, rotted-looking tissue. You should have only healthy, white tissue, with a few leaves attached. Lay it in a cool, dry area for a few days, until a dried-looking callus forms on the end.
Fill a pot -- I think I'd use a 4" pot -- with a good potting mix made specifically for cacti and succulents. Place the callused end of the stem about an inch deep into the potting mix. Keep it in bright light, but not
in direct sunlight. Wait 3 to 5 days, and then water the plant thoroughly. Give it enough water so that the water runs out the drainage holes in the bottom of the pot. Do not let the pot stand in water, i.e. if it's in a saucer or tray, pour off any water that accumulates. Don't water it again until the soil is dry about an inch down. Test it for dryness by digging into it with your fingertip. You probably won't need to water again for about a month, but test the soil every couple of weeks, just to be sure.
Ordinarily, aloes aren't so picky about the size of their pots, but in this case, since there are no roots, I recommend a smallish pot. You don't want the stem surrounded by a large amount of soil that will take a long time to dry out between waterings. It can lead to rotting of the stem. That's why you want to start the rootless stem in dry soil. Once the plant develops a healthy root ball, you could probably plant it in ordinary potting mix. If you tend to overwater even a little bit, though, it's better to use the lighter mix designed for cacti and succulents. It can prevent a lot of problems.
I don't know where you're located, so I will recommend that you keep the plant indoors. If it's windy or very hot where you live, the plant could die from dehydration outdoors, since it has no roots at the moment. Later, after it is rooted, you should be able to acclimate it to living outdoors, at least during the summer.
HTH! Let me know how the little guy fares, okay?
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams