Twenty-second Amendment to the United States Constitution
The other side of the debate argues that the 12th Amendment, in describing how elections are to be carried out, is enumerating additional requirement for holding the office of President.  In support of this side of the argument is the fact that the requirements for holding the office of President are not restricted to Article 2 (where the main requirements like age and citizenship are listed).  For example, impeachment is described in Article 1, Section 3 and upon impeachment, conviction, and removal from office a person becomes ineligible to hold the office in the future.  Similarly, the 14th Amendment establishes a requirement that a President must not have fought against the United States or given aid and comfort to its enemies. These amendments suggest a pattern of enumerating additional requirements for the presidency and proponents of this side of the debate would argue that the 22nd Amendment was intended to add yet another requirement.