You've raised a fascinating subject. I believe there is no such constitutional bar recognized presently, though many people believe there is and a vigorous minority claims that the constitutional bar has been abrogated by a conspiracy of lawyers.
When I was admitted to the bar, I was of course let in on the conspiracy. I therefore can refer you to a website that is rather enlightening on the general subject (though I am by secret oath required to reference only souces that are hostile to the lawyers-as-conspirators theory). See <<http://www.thirdamendment.com/nobility.html>>.
In a message dated 10/7/2003 8:38:05 PM Eastern Daylight Time, "Nancy Gish - Women's Studies" <[log in to unmask]> writes:
>The Supreme Court does not define authority; it interprets the
>Constitution. Like the president and the congress, the Supreme
>Court can do only what the Constitution says it can. . . . They could not, >for example, rule that an American could become a duke since that
>is forbidden in the Constitution.>>