1) A baby born abroad to one U.S. citizen parent can establish United States citizenship by proving that the citizen parent had been physically present in the U.S. or its territories for a period of at least five years at some time in his or her life prior to the birth, of which at least two years were after his or her 14th birthday.
2) Your daughter will need to register the birth and apply for proof of citizenship in order to establish citizenship. If this is not done by the child's 18th birthday, the child could lose the chance to establish U.S. citizenship.
3) If paternity is established, then both parents have equal rights to the child. The father can establish his relationship with the child by filing an Acknowledgment of Paternity (reconnaissance de paternité) declaring himself to be the child's father.
4) She may not be able to return with the baby if the father doesn't want her to. A court would have to decide whether she's allowed to return. Many airlines won't allow a parent to board an international flight with a minor child unless the parent has permission from the other parent, or a court order allowing her to do so.
5) The only benefit to the baby being born in the U.S. is easier establishment of U.S. citizenship.
6) In determining the practicalities concerning parental responsibility, the judge will take several elements into consideration, such as (Articles 373-2 and 373-11, CC):
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The agreements previously entered into by the parents.
The feelings expressed by the child.
The ability of each parent to assume their duties and to respect the rights of the other.
The results of social and psychological investigations.