Which body of government conducts elections in Texas?

The Secretary of State in Texas is the designated official responsible for conducting elections. This role includes overseeing the administration of elections, maintaining the integrity of the electoral process, ensuring compliance with election laws, and providing guidance to local election officials. The Secretary of State's office manages the election process from filing applications for candidates to certifying election results, making it a crucial entity in the state’s electoral framework.

The other entities listed have different functions; for instance, the Governor's Office is primarily concerned with the executive branch of the state government, the Texas Legislature focuses on creating laws, and the State Supreme Court handles judicial matters, including cases related to election disputes, but it does not conduct elections. Thus, the role of the Secretary of State encapsulates the responsibilities related to organizing and administering elections in Texas, making it the correct answer.