I believe that you can order a serialized replacement case, and, that the replacement case is serialized with the same number as your existing case, on an exchange basis. Meaning, that you send your existing case in along with payment, and Omega exchanges it for a new one, engraved with your original serial number. And to be clear, it is not something that you can do directly, but something that you can do through an Omega Boutique, Service Centre, AD, or Omega-licensed independent watchmaker. (And if done through a Boutique or AD, they will likely send it to the Service Center, who will actually do the exchange.).
@Archer may be able to confirm or correct my understanding.
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