While there are amendments that affect who can be president, the rule that you must be a natural born citizen is from the original text of the Constitution:

[Article II, Section 1, Clause 5]

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.