Yes its the law, and its the most mis-interpreted and rewritten of all of our laws. The 14th amendment never changed the meaning of natural born as referenced by the amendment's author John Bingham with his statement:
"I find no fault with the introductory clause, which is simply declaratory of what is written in the Constitution, that every human being born within the jurisdiction of the United States of parents not owing allegiance to any foreign sovereignty is, in the language of your Constitution itself, a natural born citizen;..." (Representative John Bingham (OH) - as found in "The Congressional Globe" - House of Representatives - 39th Congress - 1st Session - 1866 - page 1291.
Again Attorney Mario Apuzzo explains what the 14th amendment is and isn't.
Your definition of the 14th would allow Osama Bin Laden to impregnate one of his wives, ship her here to give birth, raise the child in his cave, then return the child to the U.S. as an adult, send him to college here, let him live here for 14 years, and at the age of 35 run for President. Hardly what the framers had in mind or Rep. Bingham!!!!!
It doesn’t really matter what the Framers had in mind. That may sound crass but it’s true. Parents owing allegiance to foreign sovereigns are diplomats. This has been referenced multiple times. I am not going to make the same case over and over again. As of 2010 in the United States of America you can gain citizenship in 2 ways, by becoming “Naturalized” or by being born a citizen. There is absolutely no 3rd form of citizenship. There has not been any case which has come out and said, “Bob Smith you are a citizen from birth but not a natural born citizen.”