No, Saul in the Old Testament and Paul (originally Saul) in the New Testament are different people.
* Saul (Old Testament): He was the first king of Israel, anointed by the prophet Samuel. His story is primarily found in the books of 1 Samuel.
* Paul (New Testament): Originally named Saul, he was a Jewish Pharisee who persecuted Christians. He had a dramatic conversion experience and became a prominent apostle who spread the teachings of Jesus Christ. His story is primarily found in the book of Acts and his letters (epistles) in the New Testament.
Even though Paul's Hebrew name was Saul, they are two distinct figures in the Bible.