It is important to note, contrary to what the OP says, that a baby born in the US is automatically a birthright citizen, regardless of the circumstances or the intent or any actions of the parents.
The Fourteenth Amendment says,
"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside."
Here in the Southwest, it becomes a tricky problem. We have many illegals here, a good proportion of them young women. An illegal immigrant who gives birth in the US confers full citizenship on the child. Can we be so cruel as to deport the mother of a US citizen and send the child with her?