Yes, it does say such thing. Unless, of course, your Constitution says something other than everyone elses.
“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;”
Unless Barack Obama was alive when the Constitution was adopted, he cannot be President. He is ONLY a citizen of the United States.
That does not say 'citizens of the United States’ cannot be President.
It just limits the Presidency to the subset of citizens who are natural-born. Like Barrack Obama. Or Mr. Ark (but not citizens like me, who were naturalized).
The class of citizens includes both natural born citizens and naturalized citizens. Only the former can serve as President.