Ask Mr. Smarty Plants

The black mold that you are seeing is probably sooty mold.  Sooty mold is usually caused by the honeydew excretions of sucking insects such as aphids and scale, which often suck plant juices from the protected undersides of leaves where you noticed the mold on your plant.  The mold is likely less harmful to the plant than the insects are.  Your least toxic choice for controlling scale or aphids is to release their natural predators into your yard, such as lady bugs or parasitic wasps, which are available at many plant nurseries and through mail order.  Follow the instructions that come with the insects - they should tell you to release the insects in the evening and to water beforehand to encourage them to take up residence in your yard rather than fly off to someone else's.  Your next option is to use an insecticidal soap or horticultural foliar oil, depending on whether you have aphids or a type of scale. You should only use these sprays on the infected plant, because they will also kill beneficial insects.  I have attached some links below to a similar Smarty Plants question and to a few websites that will help you determine what insect you are dealing with and what method of control will work best for you.  If these methods don't work, the plant is likely too stressed to ever do well, even if you were to use a more agressive pesticide that could harm other insects and the birds that you are trying to attract.  In that case, you may want to cut your losses and replace the diseased plant with a new, healthy specimen.  Happy bird-watching!

Control of sooty mold from aphids in Crape Myrtle

What's That Bug? - Aphids, Scale Insects, Leaf hoppers, and Tree hoppers

Texas Agricultural Extension Service - Scale insects on ornamental plants

Audubon - You have a choice!