Congratulations!

Interesting question: according to this site, http://www.multiplecitizenship.com/wscl/ws_USA.html
1) baby will receive US citizenship automatically simply by being born in the US irrespective of the citizenship of its parents.
2) multiple citizenship is permitted but not encouraged by the US government (however I don't believe there are problems when it comes to UK/US combination)

I've just had the joy of sorting out my baby's nationality:
i) Baby's nationality can be claimed through the father
ii) So far as I understand, baby needs a UK birth certificate. You can register an overseas birth at the nearest British Embassy or Consulate.
iii) After that, you can get a British passport for the baby.

Your best bet is to contact your nearest British Embassy/Consulate for advice on registering the baby's birth and claiming British nationality as a dual national for him. Otherwise the Border and Immigration Agency, or whatever it's called this week, has a nationality advice line on +44 151 672 5626. See http://www.ind.homeoffice.gov.uk/applying/nationality/ for more.

 

My mom is a Swedish citizen and lives in the US not as a citizen, but a permanent resident. My dad is a US citizen, and I was born in the US, so naturally I am too. When I turned 18, I was given the choice of having dual citizenship and a Swedish passport, and I did that. Maybe it's the same/similar with Britain?