Both our son and daughter were born in the same American Hospital in Germany just 7 years apart (1963 and 1970 when my my husband was stationed in Germany at 2 different posts. We registered our son and daughter at the American Consulate after they were born and each became an American Birth Certificate from the Consulate. I became a Naturalized Citizen on June 12, 1977 in Junction City, Kansas when my husband was stationed at Fort Riley. The ceremony was held on June 14, at the Dwight D Eisenhower center in Abilene, KS
Yes statutes do cover your children as US citizens.