Article II section 1 paragraph 5 of the U.S. Constitution states:

<< No person except a natural born citizen, or a citizen of the United
States, at the time of the adoption of this Constitution, shall be
eligible to the office of President; neither shall any person be
eligible to that office who shall not have attained to the age of
thirty five years, and been fourteen Years a resident within the
United States.>>

This is the Eligibility Clause defining the qualifications to hold
the office of the Presidency, and the phrase,"or a citizen at the time
of the adoption of the Constitution", within the Clause  is the
Exception Phrase.

    I. Why no foreigners and why the exception?

Consider a subject on which Hamilton and Jefferson, at odds over most
things, both agreed:

Thomas Jefferson:

<<Yet from such [absolute monarchies], we are to expect the greatest
number of emigrants. They will bring with them the principles of the
governments they leave, imbibed in their early youth; or if able to
throw them off, it will be in exchange for an unbounded
licentiousness, passing as is usual, from one extreme to another. It
would be a miracle were they to stop precisely at the point of
temperate liberty. Their principles with their language, they will
transmit to their children. In proportion to their numbers, they will
share with us in the legislation. They will infuse into it their
spirit, warp and bias its direction, and render it a heterogeneous,
incoherent, distracted mass.>>
("Notes on Virginia," 1782)

Alexander Hamilton:

<<The opinion advanced [by Jefferson, ] is undoubtedly correct, that
foreigners will generally be apt to bring with them attachments to the
persons they have left behind; to the country of their nativity, and
to its particular customs and manners. They will also entertain
opinions on government congenial with those under which they have
lived; or, if they should be led hither from a preference to ours, how
extremely unlikely is it that they will bring with them that temperate
love of liberty, so essential to real republicanism? There may, as to
particular individuals, and at particular times, be occasional
exceptions to these remarks, yet such is the general rule. The influx
of foreigners must, therefore, tend to produce a heterogeneous
compound; to complicate and confound public opinion; to introduce
foreign propensities. In the composition of society, the harmony of
the ingredients is all-important, and whatever tends to a discordant
intermixture must have an injurious tendency.">>
("Examinations of Jefferson's Message to Congress of December 7th,
1801," Jan. 12, 1802)

George Washington:

<<My opinion, with respect to emigration, is that except of useful
mechanics and some particular descriptions of men or professions,
there is no need of encouragement, while the policy or advantage of
its taking place in a body...may be much questioned; for, by so doing,
they retain the Language, habits, and principles (good or bad) which
they bring with them.>>
(Letter to John Adams, Nov. 15, 1794)

Alexander Hamilton suggested the following Presidential qualifications
appearing in the first draft of the Constitution:

<<No person shall be eligible to the office of President of the United
States unless he be now a Citizen of one of the States, or hereafter
be born a Citizen of the United States.>>

However, John Jay wrote in a letter to Washington:

<<Permit me to hint, whether it would be wise and seasonable to
provide a strong check to the admission of Foreigners into the
administration of our national Government; and to declare expressly
that the Commander in Chief of the American army shall not be given to
nor devolve on, any but a natural born Citizen.>>

What is the meaning of "natural born citizen"?

Benjamin Franklin in a letter to Charles W.F. Dumas, December 1775
wrote:

<< I am much obliged by the kind present you have made us of your
edition of Vattel. It came to us in good season, when the
circumstances of a rising state make it necessary frequently to
consult the /Law of Nations/. Accordingly, that copy which I kept
(after depositing one in our own public library here, and send the
other to the College of Massachusetts Bay, as you directed) has been
continually in the hands of the members of our congress, now sitting,
who are much pleased with your notes and preface, and have entertained
a high and just esteem for their author.>>

The English version of Emmerich de Vattel  1759 Laws of Nations

Chapter One, 19, section 212:

Of the Citizens and Natives. (Des Citoyens et Naturels)

<< The citizens are the members of the civil society; bound to this
society by certain duties, and subject to its authority, they equally
participate in its advantages. The natives, or natural-born citizens,
are those born in the country, of parents who are citizens. As the
society cannot exist and perpetuate itself otherwise than by the
children of the citizens, those children naturally follow the
condition of their fathers, and succeed to all their rights. The
society is supposed to desire this, in consequence of what it owes to
its own preservation; and it is presumed, as matter of course, that
each citizen, on entering into society, reserves to his children the
right of becoming members of it. The country of the fathers is
therefore that of the children; and these become true citizens merely
by their tacit consent. We shall soon see whether, on their coming to
the years of discretion, they may renounce their right, and what they
owe to the society in which they were born. I say, that, in order to
be of the country, it is necessary that a person be born of a father
who is a citizen; for, if he is born there of a foreigner, it will be
only the place of his birth, and not his country.>>

However, one must realize a mistranslation from the original French:
"Des Citoyens et Naturels": [ . . .Les Naturels, ou Indigènes, sont
ceux qui sont nés dans le pays, de Parens Citoyens . . .] "Indigènes",
or alternatively in French "originaire", is not "natives" but rather
literally in English "indigenous" as in original people over
successive generations. This was not as important then as it would
become over a century later when "native citizen" in American English
meant having been born on U.S. soil irrespective of parentage.

The Exception Phrase meant that before the time of the adoption of the
Constitution, that is before there was a United States of America, no
citizen could be a natural born citizen even though he be born on the
land that would become the U.S.A. and all such were by British law
natural-born British subjects, a condition, by English Common Law,
would could not change even by individual election. The Exception
Phrase itself proves a clear distinction between a "citizen" and a
"natural born citizen."

    II. Can one be a natural born citizen if not born on U.S. soil?

The first immigration act in 1790 prohibited citizenship to children
whose fathers never gave intent to permanently reside in the United
States and a Natural Born Citizen, was defined as being one "that may
be born beyond sea, or out of the limits of the United States,
provided that the right of citizenship shall not descend to persons
whose fathers have never been a resident in the United States:"

In legal terms, American, contrary to medieval English Common Law,
preference has been as much to jus sanguines ("right by blood", by
inheritance) as to jus soli ("right by soil", by situation or
location). However, even English Common law was revised as Britain
became a colonial power and passed a law in 1677 granting natural born
citizenship to British subjects born overseas.

    III. Can one be a natural born citizen if born on U.S. soil to
foreign parents? What about the Fourteenth Amendment and the Wong Kim
Ark  Supreme Court decision of 1898 and and later federal case-law
history?

The principle author of the 1866 Civil Rights Act and contributor to
the 1868 Fourteenth Amendment giving rights of U.S. citizenship to
freed slaves, with the intent to overturn the 1857 Dred Scott
decision, Senator John Bingham spoke in 1866 about the rights of U.S.
citizens:

<< [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, in the language of your
Constitution itself, a natural born citizen... >> [emphasis added]

At issue in the United States v.Wong Kim Ark (1898):

The 1882 Chinese Exclusion Act prohibited persons of the Chinese race
from coming into the United States or becoming naturalized U.S.
citizens. Chinese immigrants already in the U.S. were allowed to stay,
but were ineligible for naturalization. The effect of this law,
unamended, would be to create a permanent and self-perpetuating class
of people living without U.S. citizenship but also possibly over
generations without access to rights afforded by the Empire of China --
virtual stateless people.