There's no need for a constitutional amendment.
Below is the re-post of the summary and link of the excellent Breibart constitutional analysis showing that denying citizenship to the children of illegal aliens is fully consistent with the Constitution’s Fourteenth Amendment.
The law itself:
Under current immigration law—found at 8 U.S.C. § 1401(a)—a baby born on American soil to a (1) foreign ambassador, (2) head of state, or (3) foreign military prisoner is not an American citizen.
The Citizenship Clause of the Fourteenth Amendment provides: “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.”
The Civil Rights Act [1866] included a definition for national citizenship, to guarantee that former slaves would forever be free of the infamous Dred Scott decision which declared black people were not American citizens. That provision read, “All persons born in the United States, and not subject to any foreign power, excluding Indians not taxed, are hereby declared to be citizens of the United States.”
That was the original meaning of the jurisdiction language in the Fourteenth Amendment. A person who is “subject to the jurisdiction” of the United States is a person who is “not subject to any foreign power”—that is, a person who was entirely native to the United States, not the citizen or subject of any foreign government.
Under Article I, Section 8, Clause 4 of the Constitution, Congress has absolute power to make laws for immigration and for granting citizenship to foreigners. Congress’s current INA is far more generous than the Constitution requires.
In case law:
In 1884, the Supreme Court in Elk v. Wilkins noted that the language of the Civil Rights Act was condensed and rephrased in the Fourteenth Amendment and that courts can therefore look to the Civil Rights Act to understand better the meaning of the Fourteenth Amendment. The Court reasoned that if a person is a foreign citizen, then their children are likewise not constitutionally under the jurisdiction of the United States, and therefore not entitled to citizenship. In fact, the Court specifically then added that this rule is why the children of foreign ambassadors are not American citizens.
That is why Congress can specify that the children of foreign diplomats and foreign soldiers are not Americans by birth. They’re not “subject to the jurisdiction” of the United States. Congress’s INA does not grant them citizenship; federal law never has.
http://www.breitbart.com/big-government/2015/08/18/constitution-doesnt-mandate-birthright-citizenship/
The 1898 Wong Kim Ark case, on which many base their argument, is off point because baby Ark’s parents were in the U.S. legally, not illegally.
Thanks for that.