There's a constitutional law that says all people born in the U S are citizens of the U S. Period.
I understand that this is in response to another poster comparing analogies, but this reminds me of what I posted last week regarding the email-wipe / classified info / FBI - H.Clinton thread (about poor/incomplete legal reporting) - the reporting on this constitutional issue certainly leaves the impression that your statement is true, despite the fact that it is not true.
The pertinent constitutional amendment (14th Amdt's so-called "Citizenship Clause") states in relevant part:
"All persons born or naturalized in the United States, and [ii] subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." (indexing numerals not in original - added here)
As with any law (whether constitutional, statutory, or judicial precedent), it is conceptually easier to break down the law into its elements. Here, for a person to have U.S. citizenship, there are two requirements: be born / naturalized in the US, and [ii] be subject to the jurisdiction of the US.
Despite the 14th Amdt being around for 150 years, element [ii] has not been litigated and interpreted by SCOTUS. There are (at least) two divergent schools-of-thought on [ii]: one suggests that the language is surplusage and automatic upon birth on US soil (but this violates most of the cannons of statutory construction); while the other suggests that [ii] being "subject to the jurisdiction of the US" is not possible for parents in the US illegally, and therefore, any offspring born here are similarly illegal and not subject to jurisdiction (the corollary being that the illegal parents and offspring are subject to the jurisdiction where the parents are permanently and legally domiciled).
I think the latter is the more appropriate interpretation, but who knows when SCOTUS will actually entertain such a case and make a decision.
What gets reported as "law" is generally policies that have been adopted (and embodied in executive orders) that suggest the offspring of illegals born in the US are US citizens, but may (under certain circumstances) be deported until the US born child turns 18 (and who may then sponsor the parents for immigration and naturalization at 21). While policies may be implemented, policies are down the food-chain with regard to "force-of-law" questions, below regulations implemented and enforced by executive branch agencies.
One case that is getting a lot of attention in this area is United States v. Wong Kim Ark (1898), which held that Ark, a child born in the United States to legal resident Chinese immigrants, was a birthright U.S. citizen under the 14th Amendment. Dictum (statements not essential to deciding a case) from that opinion have been teased into the policies noted above. However, the Ark-decision did not address element [ii] since Ark's parents were legal residents (and thus subject to US jurisdiction).
Time will tell whether [ii] is interpreted in a way that continues (and codifies) current policy(ies) or whether it forces a contraction of current policy(ies).