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(This is from the US Department of State Website)

Acquisition of U.S. Citizenship By a Child Born Abroad

Birth Abroad to Two U.S. Citizen Parents in Wedlock: A child born abroad to two U.S. citizen parents acquires U.S. citizenship at birth under section 301(c) of the Immigration and Nationality Act (INA). One of the parents MUST have resided in the U.S. prior to the child’s birth. No specific period of time for such prior residence is required.
Birth Abroad to One Citizen and One Alien Parent in Wedlock: A child born abroad to one U.S. citizen parent and one alien parent acquires U.S. citizenship at birth under Section 301(g) INA provided the citizen parent was physically present in the U.S. for the time period required by the law applicable at the time of the child’s birth. (For birth on or after November 14, 1986, a period of five years physical presence, two after the age of fourteen is required. For birth between December 24, 1952 and November 13, 1986, a period of ten years, five after the age of fourteen are required for physical presence in the U.S. to transmit U.S. citizenship to the child.
Birth Abroad Out-of-Wedlock to a U.S. Citizen Father: A child born abroad out-of-wedlock to a U.S. citizen father may acquire U.S. citizenship under Section 301(g) INA, as made applicable by Section 309(a) INA provided:
1) a blood relationship between the applicant and the father is established by clear and convincing evidence;
2) the father had the nationality of the United States at the time of the applicant’s birth;
3) the father (unless deceased) has agreed in writing to provide financial support for the person until the applicant reaches the age of 18 years, and
4) while the person is under the age of 18 years —
A) applicant is legitimated under the law of their residence or domicile,
B) father acknowledges paternity of the person in writing under oath, or
C) the paternity of the applicant is established by adjudication court.
Birth Abroad Out-of-Wedlock to a U.S. Citizen Mother: A child born abroad out-of-wedlock to a U.S. citizen mother may acquire U.S. citizenship under Section 301(g) INA, as made applicable by Section 309(c) INA if the mother was a U.S. citizen at the time of the child’s birth, and if the mother had previously been physically present in the United States or one of its outlying possessions for a continuous period of one year.