Unless Texas law is very unusual, he will be released when he sentence expires, even though the appeal is still pending. If his conviction is subsequently reversed on appeal he would presumably be discharged from parole (or whatever the Texas equivalent might be) and he might (depending on Texas law and on whatever record he has already) be able to get his arrest and conviction records expunged. Wait for an answer from a Texas attorney.
Mr. Sachs got it right. The defendant will be released by the authority holding him upon serving the entire sentence. The appeal will continue and is not considered moot. Trial courts differ in their policy regarding appeal bonds and your trial judge may or may not require one., but regardless of the policy, the defendant should be released upon serving the maximum sentence.