I have a new small dog, about one year old, and three existing dogs. The new dog wees on his food bowl every time he has finished a meal, whether he is fed alone or with the other dogs. Can anyone tell me why he does this.

Where did you get him from and how much do you know about his former living conditions? How long have you had him? Is he intact?

Does he do it while there is still food in the bowl or only after he has finished it?

It sounds like he is marking it as his. Does he do this to anything else that you have found so far like toys or bedding? Does he mark the water bowl, too?

The only solution I can think of is to take the bowl up before he gets a chance to pee. Maybe even while there is a bite or two left (if he does it after the bowl is empty).

I agree, it's more then likely a territorial thing. If he came from a home where he was a single dog then, he he is marking his territory. If he came from multi dog home, or living situation, he is making claim on the "alpha" dog.
Does he have his own food bowl or do all the dogs share? Is he free fed or do they have specific feeding times?
If you are free feeding, I would suggest specific feeding times, (usually the healthiest method to begin with), if they share, I would suggest getting them their own bowles. If they all have different bowls, and specific feeding times, then I would try what cat1864 suggested with removing the bowl as soon as he is finished.