Texas Local Government Code - LOC GOV'T § 23.002. Filling Vacancy in Municipal Office

(a) The aldermen on the governing body of the municipality shall fill any vacancy that occurs in an office created by this chapter or created under this chapter by the governing body unless an election to fill the vacancy is required by Article XI, Section 11, of the Texas Constitution. The vacant office shall be filled for the unexpired term only.

(b) A person serving as a member of the governing body is not, because of that service, ineligible to be appointed to fill a vacancy in the office of mayor of the municipality, but the person may not vote on the person's own appointment.