It will help if you identify the country of the father for us.
Here are the laws for the US
Acquisition of U.S. Citizenship by a Child Born Abroad
As you will see, if the couple is NOT married, the baby will have US citizenship ONLY if the father agrees to support her until she is 18.
As for the mother, I do not believe that having a child of an American brings her any special rights unless the father brings her over as a wife, supports her, etc. OR the child does when she can support her after 18.
The child has to be acknowledged by the father and registered at the US Embassy here
Acquisition of U.S. Citizenship by a Child Born Abroad
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) of the INA provided the U.S. citizen parent was physically present in the United States or one of its outlying possessions 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, is required for physical presence in the United States or one of its outlying possessions to transmit U.S. citizenship to the child.) The U.S. citizen parent must be the genetic or the gestational parent and the legal parent of the child under local law at the time and place of the child?s birth to transmit U.S. citizenship.
Birth Abroad Out-of-Wedlock to a U.S. Citizen Father ? ?New? Section 309(a)
A person born abroad out-of-wedlock to a U.S. citizen father may acquire U.S. citizenship under Section 301(g) of the INA, as made applicable by the ?new? Section 309(a) of the INA provided:
A blood relationship between the person and the father is established by clear and convincing evidence;
The father had the nationality of the United States at the time of the person?s birth;
The father was physically present in the United States or its outlying possessions prior to the child?s birth for five years, at least two of which were after reaching the age of 14.
The father (unless deceased) has agreed in writing to provide financial support for the person until the person reaches the age of 18 years, and
While the person is under the age of 18 years --
the person is legitimated under the law of his/her residence or domicile,
the father acknowledges paternity of the person in writing under oath, or
the paternity of the person is established by adjudication of a competent court.
Birth Abroad Out-of-Wedlock to a U.S. Citizen Father ? ?Old? Section 309(a) of the INA- A child born out-of-wedlock to a U.S. citizen father may acquire U.S. citizenship under the former Section 301(a)(7) of the INA as made applicable by the ?old? Section 309(a) of the INA if the U.S. citizen father, prior to the child?s birth, had been physically present in the United States or one of its outlying possessions for ten years, five of which were after the age of 14, and if the paternity of the child had been established by legitimation prior to the child reaching the age of 21. The ?old? Section 309(a) of the INA is applicable to individuals who were 18 on November 14, 1986 and to individuals whose paternity had been established by legitimation prior to that date. Individuals who were at least 15 on November 14, 1986, but under the age of 18, could opt to have their claim determined in accordance with the provisions of either the ?old? or the ?new? Section 309(a).
