Texas Special District Local Laws Code - SDLL § 3861.054. Vacancy

(a) If a vacancy occurs in the office of an appointed director, the mayor and members of the governing body of the city shall fill the vacancy for the remainder of the director's unexpired term in the same manner as the original appointment.

(b) If a vacancy occurs in the office of an elected director, the remaining directors shall fill the vacancy by appointing a person who meets the qualifications prescribed by Section 3861.053. If there are fewer than three directors, the mayor and members of the governing body of the city shall appoint the necessary number of directors who meet the qualifications prescribed by Section 3861.053 to fill all board vacancies, regardless of whether the vacating directors were appointed or elected.