Fabrics that fray so quickly and completely that they'll practically disappear on you typically benefit from an enclosed seam. The most common enclosed seam is a bias tape bound seam. this type of seam finish works well for easily frayed fabrics, fur, and unlined jackets that you need to give a professional look to. It's also fairly easy to pull off if you have some coordinating bias tape lying around. If you don't, head off to your local store and buy some. Make sure you've measured your seams and added up how much you'll need so you don't need to make a second run.
Now that you have your bias tape, it's time to enclose and finish your seam. Start by pressing your seam open. Then encase each raw edge with the bias tape, pinning carefully so you'll catch both sides of the tape and the fabric when you sew the seam. Stitch close to the folded edge of the bias tape, making sure you're not missing the tape on the other side. Once you stitch both sides of the seam, you're finished.
You can make your own bias tape, if you like. This is done by simply cutting long strips of fabric and folding and pressing so that it is folded just like bias tape. Examine a piece of bias tape to see how it's folded and you'll discover making your own out of whatever fabric you wish isn't hard.
Whether you make it or buy it, make sure to wash and dry the bias tape before you use it. Nothing is more irritating than having your bias tape shrink once you've already finished your garment. So always, always pre-shrink your bias tape.