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I believe that will take a Constitutional amendment.

Nope. As written, the 14th Amendment was NOT intended to grant citizenship to the children of foreign subjects.

The Slaughterhouse Cases are the first Supreme Court interpretation of the 14th Amendment on record. The author of the majority opinion is a contemporary of those who drafted and debated the Amendment. The following text is from the majority opinion (about 3/4 of the way down the linked source page):

http://supct.law.cornell.edu/supct/search/display.html?terms=Slaughterhouse%20Cases&url=/supct/html/historics/USSC_CR_0083_0036_ZO.html

Slaughterhouse Cases, 83 U.S. 36 (1872) (USSC+)
Opinions
MILLER, J., Opinion of the Court

"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 first observation we have to make on this clause is that it puts at rest both the questions which we stated to have been the subject of differences of opinion. It declares that persons may be citizens of the United States without regard to their citizenship of a particular State, and it overturns the Dred Scott decision by making all persons born within the United States and subject to its jurisdiction citizens of the United States. That its main purpose was to establish the citizenship of the negro can admit of no doubt. The phrase, "subject to its jurisdiction" was intended to exclude from its operation children of ministers, consuls, and citizens or subjects of foreign States born within the United States.

Here is a second source:

"Every Person born within the limits of the United States, and subject to their jurisdiction, is by virtue of natural law and national law a citizen of the United States. This will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States, but will include every other class of persons.

Senator Jacob Howard, Co-author of the citizenship clause of the 14th Amendment, 1866.

Senator Howard recognized three classes of people to whom the 14th Amendment citizenship clause would not apply: foreigners (tourists here temporarily), aliens (those here illegally but who have no intention of leaving), and foreign diplomats (here legally and in a special protected status who will leave upon the expiration of their term).

And in Section 5 "The Congress shall have power to enforce, by appropriate legislation, the provisions of this article." cedes control of implementing provisions of the Amendment back to Congress. Because the Constitution is a limiting document they MAY NOT grant citizenship to illegals, nor the equivalent.

The interpretation started to change when a Roosevelt court started playing fast and loose with the term, "subject to the jurisdiction thereof" in Bridges v. Wixon (1945).