Azaleas can be propagated in four ways: seeds, stem cuttings, layers, which is a variant of stem cutting involving cutting and bending branches to form new plants, and by grafting a cutting onto the root system of another plant. When properly prepared, a broken azalea branch can be treated like a stem cutting or layering, for the purposes of propagation.
Evergreen azaleas take very well to cutting propagation, although this technique is by no means limited only to evergreens. If the broken branch is from an otherwise healthy azalea and can be trimmed to between 3 and 6 inches long, it is possible to propagate from the branch. Remove any leaves from the lower third of the branch and plant it 3 inches deep in a moist mixture of peat and perlite.
In order for the branch to sprout roots, it must be encouraged with rooting hormone before it is planted. The rooting hormone encourages root growth where there would otherwise be none and thus ensures that the broken branch can form the stem of a new plant. Simply planting the branch in soil and watering it will not encourage the plant to grow.
The broken azalea branch must not have dried out if it is to propagate. Dry branches are either dead or dying and no amount of rooting hormone will save them. Always try to ensure you use branches that have only recently been broken off when attempting to propagate your azaleas.