usa – Baby with an American father, traveling to America

I am a US citizen, I live in Australia, with my wife (Australian citizen) and a girl (born here in Australia).

We are traveling to America in a few weeks. We have obtained an Australian passport for my baby, hoping to use it when I enter the USA. UU

However, my understanding is that

  1. Any child born to a US father is automatically a US citizen (at least until he is 18 years of age) and
  2. America requires that you use your American passport to enter the country, if you are a US citizen.

Does this combination mean that I must register my daughter at the US consulate here in Australia and obtain a US passport for her before I can travel to the United States with her?