There are a couple of mimosa-like pests in the Southeast, but the worst, and the one you may have, is chamberbitters (Phyllanthus urinaria). That's the one with the seeds underneath the sets of leaves. It's a real nasty, and if that's what it is, you're going to need all the luck I can wish you. Take a look at the attached page from Clemson, and then google it for more information. There apparently is a herbicide that works on it, but it's not Round-up, so you may have to look for it. Good luck.

Here is a link that might be useful: Clemson page on chamberbitters

Agreed. The good news is that they come up easily. Breaking the cycle is not difficult on this one, but does require eliminating as soon as you see it because you'll never see any flowers. You may be able to get control of it with a weed-wacker. Stepping on it might break the stems, if you need more time to get to it all.

Yes I have found out it is the dreaded chamber bitters. The good news is that it pulls out easy and is a summer annual. And deep mulch helps since the seed needs light to germinate. My plan is to keep pulling and to lay down thick layers of newspaper and then re-mulch heavily. Just had a new St. Augustine lawn put down so will pay close attention to pulling any new weed I see.
Thanks for the info.