There is no required time frame. Just be sure your divorce has been finalized. This can be determined by checking the court docket or by receipt of your certified decree. You may need a copy of the certified decree when you get your new license.
It is my understanding that there is no waiting period in Ohio. However, you should check with an Ohio practitioner to make sure. Most states don't have a remarriage waiting period after divorce, but some states allow a judge to create a waiting period for the parties. If you marry before the imposed waiting period ends, your new marriage would generally be voidable rather than void. This means either party to the new marriage could ask to have the marriage declared invalid.
In some states, both parties must be given time to appeal the divorce decree before you can remarry. Those states that have remarriage waiting periods generally calculate the waiting period from the day the judge approves the divorce decree. Your final hearing date isn't necessarily the date of your divorce decree - in most states your divorce isn't final until the order is signed by the judge.
This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Laws vary widely from state to state. You should rely only on the advice given to you during a personal consultation by a local attorney who is thoroughly familiar with state laws and the area of practice in which your concern lies.
Sign up to receive a 10-part series of useful information and legal advice about the divorce process.