That was fast … Nicole Kidman and Keith Urban are officially single in the eyes of the law … because their divorce has been finalized.

According to new legal docs, obtained by TMZ, a judge in Tennessee on Tuesday signed off on the couple’s divorce.

We broke the story … Nicole filed for divorce back in September, citing irreconcilable differences as the reason for the split.

Nicole and Keith had already hammered out all the divorce details before Nicole filed … so they did the leg work early on and were just waiting for the judge’s signature.

As we’ve told you … Nicole and Keith waived any spousal and child support, and they each retain their own assets, which suggests a prenuptial agreement.

They agreed to a parenting plan and co-parenting classes … the parenting plan says the kids spend 306 days with Nicole and 59 days with Keith. They also agreed that all major decisions involving their kids, including education and health care, will be made jointly.

As we first reported … Nicole and Keith separated in June, and we were told she wanted to save the marriage, although by all accounts, Keith did not.