Quite a lot of the above answers are wrong.
You do not have to go to the area where the car is registered to change ownership/plates. That can be done at the buyer's local Land Transport Office. I did exactly that last year.
Secondly, a couple of years ago I was told that if a car is registered in Bangkok, it does not have to have new plates, the number can stay but the owners address will change to their current address. I was told that this only applies to cars with a Bangkok registration - all others must be given a new number.