@Walter Hinteler,
If the child has an American father, the child is considered an American citizen. Albeit, a child with duel citizenship. One of my sons friends became enamored with a German women when they both worked in Bermuda. They share custody now of two children, however, the mother returned to Germany, the father followed and now they are estranged but the father has remained in Germany in order to have access to his children. I've met the children, they are very attractive but they are incredibly biased against Americans. I've worked with many families from around the world but I've never met children who are so disrespectful around adults. Please let me emphasize that I've been around many Eastern European dependent children, but these two are a glaring exception. I assume it's because the parents are estranged.
I'm sure your old flame is a terrific mother, a wonderful citizen et al. But get a lawyer who understands international law and international parental issues. Germany will protect the child as a German citizen, and the US will see the child as an American citizen. So you are stuck with this uncertain situation. I understand you want to believe this child is yours, however, she might not be. You truly must get a DNA test and hopefully there is an American Consulate that can oversee the test. I commend you for taking responsibility, however I caution you do take every step to make sure this child is yours.
If you want this child to be yours, I truly hope she is. However, there are many legal issues involved, including inheritance. Might not sound like a big deal now, but if you die before your parents, or die before a legal spouse and children do, this child in Germany is entitled to a portion of your estate. If she is your child, she deserves to be an heir, if she isn't, this woman could be stealing from your future children. I'm sure right now you are an emotional wreck, I would be, that's why you need to take a step back and touch all the legal buttons. All I can tell you is please do not get attached before you find out for sure this child is your mothers grand daughter. If you don't, there will always be a possibility your entire family will be devastated if after years of thinking this child is family it's discovered she's not.
Actually my biggest concern is the baby. She deserves an honest account of who her parents are and everything that goes along with it. Especially, the medical history. Which country gets to claim the child as a citizen pales in comparison to the rights of this little girl knowing who her father is. It's pretty easy to determine who mom is, it's tricky without DNA to determine who dad is.