No matter what type of fence you have or how well constructed it is, if your dog wants to dig under it, he will. Unless you've constructed an actual dog-run area with a concrete floor, it's tough to deter a determined dog.
There is, however, a simple solution. A 24-inch-wide strip of wire fencing, laid on the ground and attached to the fence posts, will work wonders. It's inexpensive, safe for the dog, and can be adapted to virtually any type of existing fence. The dog doesn't like the feel of the wire underfoot, and usually won't walk on it. He can't dig through it, and if he starts a hole in front of the strip, the wire will sag down into the hole and impede further progress.
You can purchase a variety of types of light-duty fencing at most hardware stores and lumber yards. If possible, purchase 24-inch-high fencing to save having to cut it. Otherwise, use 48-inch fencing and cut it in half with wire cutters or snips. You might also check with any local companies that install chain link fences--they may have narrow remnants from past jobs, and the flexible chain link fabric is ideal for this application.
Measure the perimeter of the fence to determine how much fencing you'll need, allowing enough to overlap at the corners. You'll also need some heavy screw eyes--one for each post in the existing fence, and a small coil of baling wire. If your fence has metal posts, you can skip the screw eyes.
Begin by drilling or punching a small hole in each fence post, about an inch up from the ground, then install a screw eye securely in each hole. Next, lay the fencing along the perimeter of the existing fence, just touching the posts. Using short lengths of wire and a pair of pliers, secure the fencing to the screw eyes and you're finished. For metal posts, just wire the fencing strip directly to the posts.
If you've got another hour to spare, you can dog-proof the gate, too. Open the gate, measure the distance between the two supporting posts and cut two pieces of pressure-treated 2-by-6 lumber to that length. Using galvanized nails, nail the boards together to form an "L."
Dig a 6-inch-deep by 6-inch-wide trench between the gate posts. Install the 2-by-6 L so that the side leg faces down and into your yard and the top leg is about 1 inch above the ground. Level the board, then toenail the top 2-by-6 to the gate posts. Finally, backfill the trench with dirt and gravel.