We solved this problem by attaching an apron around the inside perimeter of the fence, much as Garrett describes above. The apron consisted of 2' woven wire fencing. Thus, the apron will prevent digging within two feet of the fence. Animals are not smart enough to dig two feet out and tunnel under the apron. They will always run all the way to the fence and attempt to dig.
We rolled out the wire mesh parallel to the fence and secured it flat with landscaping pins. The apron is then attached to the chain link using hog rings. Hog rings are quick and easy to attach, but they require a special set of hog ring pliers (not expensive).
Afterward, cover the apron with plenty of soil. Eventually the grass will cover it back up. If you covered grass with the apron, the grass will gradually grow up through the extra soil covering the apron. You'll forget it is there.
But a determined dog can chew through the chain link (ask me how I know this). In that case, an electric fence is needed. We put in about 500' of electric fencing - one strand at about 12" to 18" off the ground. Our dogs learned the first day to stay away from it. So did I. The cost was around $100.
An electric fence is much easier to install than an "invisible fence". It's also cheaper and more reliable. An invisible fence wire has to form a loop. It uses light-gauge wire (18-gauge) which is cheap, but vulnerable to breaks (and good luck finding the breaks - good luck repairing them). But an electric fence runs from the powered terminal of the energizer out to wherever your fence ends. The circuit is not closed until something or someone touches the wire and becomes the path to ground. And since it is visible, it's easy to find any damage.
The disadvantage is that it affects any animal or human that touches it when it's powered up. The invisible fence only affects an animal wearing a collar (but the collar itself is expensive and the batteries are an ongoing expense).
The electrified fence is also visible and not terribly attractive and might not be allowed in some neighborhoods.