Neighbors chain link fence doesn't touch the ground?

Not sure how long the fence is, But you could pick up some 5/4 pressure treated decking and some pickets (The vertical 2x2s for deck railings) The 5/4 is about 5 1/2 inches wide and you could stand it up on edge against his fence and use the pickets as stakes to hold it there then just add some screws and you have a long lasting barrier.

Additional

Please don't use chicken wire or any other type of wire. Your dog will still think he can get through it and will try to dig. This can severely damage his paw pads and cut into his gums if he tries to bite at it. putting up a wooden barrier will stop him from thinking this way and it is much safer for him. I have 3 small dogs and I wouldn't listen to people who obviously never had to care for a loved pet.

I suggest you install your pretty fence 6 inches inside the boundary line (no, you're not giving away the land on the other side -- you still own it). If the neighbor is a jerk, it's a good idea to put the second fence inside instead of on the property line. Set fence posts in concrete, always. You can rent a power auger (post hole digger) locally. Be careful not to electrocute yourself. Have the utility companies flag their lines for you (no charge), and be sure you pull a building permit. The permit officer can be a big help to people who are new at this stuff.

The chain link is okay. It makes it easier to trim grass. Ask your neighbor if you can put down some chicken wire on your side (over lap it on the ground) to prevent your pup from getting out. If you extend it on to the ground, it will be hard for your pet to go under. If it finds a spot to lift it up, place a piece of lumber or heavy rocks every 2-3 feet so it can't lift it up.

Put a short fence in front of their fence, close enough so the dog can't get under their fence. Keeping your dog out of their yard is your responsibility anyway, not theirs.

The fence is not improperly installed, most chain link fences are an inch or two from the ground. You have to deal with the problem, being your dog not staying in your yard. Tie him up or install your own fence to keep him in. Your dog and your responsibility . Not your neighbors. They should not have to go to extremes to keep your rat out of their yard

Not your fence. Not your problem.

If you want to contain your dog, figure something that you are willing to do, without involving his fence or him.

You can't ..LEGALLY...Without his permission.