[andyjohnson0]

Texas Constitution. Article 1, Section 4:

"Sec. 4. RELIGIOUS TESTS. No religious test shall ever be required as a qualification to any office, or public trust, in this State; nor shall any one be excluded from holding office on account of his religious sentiments, provided he acknowledge the existence of a Supreme Being." [1]

This doesn't actually say that an atheist can't hold office, but a non-believer can be excluded from office.

There are some particularly egregious examples of public-office tests listed in [2]; in particular Tennessee.

[1] http://www.statutes.legis.state.tx.us/Docs/CN/htm/CN.1.htm

[2] http://en.wikipedia.org/wiki/Discrimination_against_atheists...