Thanks for your question.
There is no longer a requirement to apply for a Certificate of Approval from the UKBA in order to marry. The decision of whether to marry is left entirely to the local registrar.
In most cases provided you can prove your identity, address and put the appropriate notice of your marriage up at your local registry office the registrar will agree to marry you.
In terms of whether you will then be able to immigration to the US you would have to take advice from a US immigration attorney. You have submitted your question to the UK law section and I am a UK lawyer.
You should submit a further additional question to the US law asking for advice on whether you, as an illegal immigrant, would have a prospect of a successful visa application on the basis of your marriage to a US citizen.
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