In a Texas common law marriage, with ceremony but no license, will a license alone make the marriage "legal"?

In other words, say you get married in a private ceremony in Texas but did not get a license, and meet all the legal requirements for a marriage license and later on decide to get a marriage license to make it "legal", do you have to go though another wedding ceremony before you are "legally married" according to statutory law? I am aware that common Iaw marriages are by definition legal by common law, but can't find out if statutory legality requires a repeat of public vows as well as a license.