Don't despair, fortunately aloes, like most succulents, root very readily. The best way forward is to (A) take a leaf cutting and (B) plant up the broken stem, as follows:
Remove a healthy leaf from the stem (with a sharp knife) and leave it to dry out for two or three days, until a thin 'skin' forms over the open edge;
Insert it firmly in some moist potting soil, to a depth about one third of its length;
Place the pot in a bright but sunless place;
Water sparingly - keep the soil moist but not wet;
- Repeat the last three steps using the broken stem, inserting it to a depth of 3-4".
Provided you don't over-water, your cuttings should take root by mid-September, and you will have two 'new' plants by next Spring.