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The marriage is performed at the Standesamt, which can be loosely translated as the registry/clerk's office. It is a civil ceremony and that is the ceremony that is binding. Any religious ceremony (i.e. wedding) is optional and does not affect the matter.
You will need:
-a birth certificate
-your US passport and documentation of your stay in Germany
-the Ehefähigkeitszeugnis, which is a "certificate of no impediment" that you have nothing hindering your ability to marry, such as a prior marriage that is still ongoing. US does not issue such certificates, but the US consulate can help issue a letter stating that no such certificate exists in USA, and that normally is enough.
-f you were married before, you need proof that all previous marriages were dissolved.
Marriage in Germany is a very quick affair, done civilly, before the registrar. Expect it to be over in under 30 minutes. You would need to call the local Standesamt ahead of time to make an appointment.
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