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The state of Texas did not start keeping official birth records until 1903. There is the possibility that Falls County may have some type of birth record on file. Check with the county courthouse there. You could also contact the Texas State Library and Archives Commission: https://www.tsl.texas.gov/arc/genfirst.html. They will be able to guide you to genealogy resources such as church and baptism records, which are helpful if no birth record exists.