Government Code §573.081

(a) An individual who violates Subchapter C or Section 573.062 (b) shall be removed from the individual's position. The removal must be made in accordance with the removal provisions in the constitution of this state, if applicable. If a provision of the constitution does not govern the removal, the removal must be by a quo warranto proceeding. (b) A removal from a position shall be made immediately and summarily by the original appointing authority if a criminal conviction against the appointee for a violation of Subchapter C or Section 573.062 (b) becomes final. If the removal is not made within 30 days after the date the conviction becomes final, the individual holding the position may be removed under Subsection (a).