Excellent to know about not needing to translate that paper.
Take the paper you have from the Mairie / hospital to the British Consulate in Bordeaux (first link) where you can register the birth - you'll need to fill out forms and no doubt pay a fee as per the second link below. Dad may have to do all this as he's the British citizen plus he will need all his own papers and certificates to prove he's British :-) May need all your papers too!
http://www.fco.gov.uk/en/travel-and-living-abroad/find-an-embassy/e...
http://ukinfrance.fco.gov.uk/en/help-for-british-nationals/living-i...
In Bordeaux they should also give you all the advice needed on how to apply for a passport for the baby too.
Your baby can't have a french birth certificate as she isn't French, what you have is just a statement of birth. You are going to need to go to the consulate (american or british) with the piece of paper and your ID. Get your own birth certificates ready too, as they will probably want to see them. Bring in all the documents you can think of, including justification of residence in France, as they always want something they think you don't have....
Hi there , both my children were born in France , but have UK passports , you can get a copy of your childs birth certificate by calling in to the Mayors office in the canton where your child was born, they will give you a photocopy stamped by the clerk which is considered an official copy of the extract de naissance . I've never had to get translations of this for a UK passport , the passport office seem to be fine with just that piece of paper.