Over the past few years since I met and married a British lass the subject of dual citizenship has come up several times. Speaking with several other ex-pats over those years it seems that there's a lot of misunderstanding regarding the concept of dual citizenship. It was to the extent that my wife was reluctant to become a US citizen because she thought it meant she'd have to relinquish or forfeit her British citizenship.
After years of hearing conflicting stories I decided to do some research on the subject. What I learned is that the US has no problem with dual citizenship provided you were already a citizen of another country when you became a US citizen. Where the US government DOES take issue with dual citizenship is if you're born an American citizen and decide you'd like to become a citizen of another country. Our government may ask you to relinquish/forfeit your US Citizenship upon becoming a citizen of your new country. Basically my wife can become a dual citizen but if I wanted to become a citizen of another country I might or might not have to forfeit my US citizenship to do so (I'm guessing this is somewhat contingent on what country I'd want to become a citizen of).
The root of it is, as far as the US government is concerned if you become a naturalized citizen it's your origin nation that determines whether or not you can retain your citizenship in that country-- not your adopted one, the United States. Provided the UK government takes no issue with it, my wife can become a dual citizen. For that matter my wife may already have dual-citizenship via maternal descendancy as her mother is a Kiwi.
Upon further research I learned that my daughter already is a dual citizen by virtue of having a British mother and an American father (not to mention being born in the United States). There are no special fees or hoops to jump through for her to be considered a British citizen. She can apply for a UK and/or a US passport (although I've read the US government requests that dual citizens residing in the US use a US passport upon exiting and re-entering the country-- so even if we did get her a British passport she'd also need a US Passport).