The baby IS a US citizen, and has no other citizenship. Being born in the Bahamas does not make one a Bahamian citizen. Being born to US citizen parents (even just one) DOES make one a US citizen.
Person (18 years or older), born in the Bahamas, but whose parents were not citizens of the Bahamas. Person (18 years or older), born in wedlock outside the Bahamas to a Bahamian mother. Child adopted by Bahamian citizens.