You don't apply for "dual citizenship", nor do you worry about whether the US allows/recognizes dual citizenship. You attain citizenship in each country separately according to that country's laws.
Each country makes its own citizenship laws, usually without much thought to the citizenship laws of other countries. US law determines whether or not a person is a US citizen. Since your baby was born in the US, the baby is a US Citizen according to the 14th amendment, regardless of what other citizenships she may have. The US doesn't get involved one way or another with the issue of whether or not she's a UK citizen -- that's a matter between the UK and the baby. But for US legal purposes, the child is a US citizen, and any other citizenships are irrelevant. The US won't, for example, let the UK embassy intercede on her behalf if she's ever arrested for a crime in the US. In that sense, the US doesn't "recognize" her UK citizenship, but the US doesn't mind at all if the UK considers her to be a UK citizen.
Though many countries (like the US) don't consider other citizenships in their own citizenship laws, there are some countries that DO consider other citizenships in their own citizenship laws. Outer Fredonia might say that any Outer Fredonian who naturalizes to another citizenship automatically relinquishes their Outer Fredonian citizenship. Or North Bintu might say that a child born of North Bintuan parents outside the country only inherits North Bintuan citizenship if that child did NOT get citizenship based on the country in which the birth took place. Stagswanna might say that a person may only naturalize to Stagswannian citizenship if their previous citizenship(s) are fully relinquished. There are many complexities possible.
I don't know enough about UK citizenship laws to know if being born of a UK parent is enough to pass on citizenship, or if there are other conditions which must be met. I don't even know if the UK cares that your baby is a US citizen when it determines whether she's a UK citizen (I think not, but...). Consult with the UK consulate to find out. But it might be easier to find information, espcially on a web search, if you only inquire about UK citizenship, not "dual citizenship".
See the
Dual Citizenship FAQ for more details on the issues involved.