If american woman gives birth to her child in another country. Is child still american??PLEASE Read more info.?

If the the mother is an American citizen, then the child is an American citizen. See INA Act 320. The mother will have to prove to have at least 10 years legal residency inside the United States or American territory, or a US Military Base. There were some additions made to it with the Child Citizen Act of 2001 as well.