The child support order stays in place until modified. It is simple to modify when the parents marry - just file the Motion to Modify based on changed circumstances of the child, parents now married and under same roof, and submit the agreed order.
The backpay stays in place. You might be able to get the OAG office to lower it or get a judicial finding that says the total arrearage is zero, but both require action on your part. Good luck and congrats!
Disclaimer: This answer is intended for informational and educational purposes only and should not be considered legal advice nor forming the attorney-client relationship.
He will need to file a motion to reduce child support to zero. In support, he would show that you two are now married. He will still need to pay any arrearage, but the money will come right back to the two of you. You just need to wipe clean the OAG's books on the matter.
So the exact protocol is this:
1. He files a motion to reduce child support to zero, showing in support that the two of you are married.
2. You sign a waiver of service in front of a notary and file it with the court. That means that you can now ask the judge to stop the child support.
3. Write up an agreed order FINDING that the two of you are now married and ordering that child support payments are terminated as of the date he filed the motion in step 1. Your order will have a signature line for the judge and below it should have a line that says "Agreed as to both form and substance" and then a signature line for each of you.
4. You both sign the order (does not have to be notarized).
5. Go to the district clerk's office and ask for your file one morning so you can take it to court. Get the file (bring 3 copies of your final order), and go to the bailiff. Tell the bailiff you have an agreed order and ask if the bailiff will present the order to the judge for you. He may just take it back there, get it signed, and you're done with the judge. If not, he may tell you to set a hearing date/time (which is just an appointment to talk to the judge) and come back another time.
6. Once the judge signs the order, take it back to the district clerk's office and fill out a "child support setup form"--which varies from county to county. Also fill out what we call an "Austin Form". Turn all that into the district clerk and you should be done.
It will be easier on you if you hire an attorney, but this is a really low risk proceeding. If you mess it up, the court will tell you what you did wrong and you can try again. In the mean time, you'll be paying child support to yourself, so that's why I call it a low-risk proceeding.