I looked into doing this, and I was advised that you will need more than one ring to create another ring. Mainly because there is filler in your gold, so you will need more gold added (other old jewelry, etc.) Also beware that some places will take your gold and won't actually use that gold to create your new piece. They will use the equivalent weight in new gold.
So, if it is important for your gold to go back into your ring, make sure you ask specifically.
I've never heard of this, it sounds interesting. Usually I hear about getting heirloom rings resized, either larger or smaller, or getting stones reset, but I have not heard of a new ring being made with the old metal of another ring. Do you know someone who has had it done? Have you looked for craftspeople on Etsy who might offer this service, if a traditional jewelry store doesn't provide it? A jewelry store may also have contact info for specialty craftspeople.