Tex. Local Government Code § 26.045

Sec. 26.045. FILLING VACANCY ON GOVERNING BODY OF MUNICIPALITY WITH POPULATION OF 1.5 MILLION OR MORE. (a) Except as provided by Subsection (b), if a vacancy occurs on the governing body of a municipality with a population of 1.5 million or more and more than 270 days remain before the date of the next general election of members of the governing body, the governing body shall order a special election in the district in which the vacancy occurred, or in the entire municipality if the vacancy occurred in an at-large position, to fill the vacancy. The special election shall be held on an authorized uniform election date prescribed by the Election Code that occurs before the general election and that allows enough time to hold the election in the manner required by law and shall be conducted in the same manner as the municipality's general election except as provided by provisions of the Election Code applicable to special elections to fill vacancies.

(b) This section does not apply to a municipality that has provided by charter or charter amendment a different procedure to fill a vacancy on its governing body for which the unexpired term is 12 months or less.