I don't have my weed book handy, but I'm going to guess that it's Mimosa strigillosa. There's another ground cover mimosa, I think...but I can't recall it right now.

Rhizo - you got it! Thanks.

I have two different weed killers. I have a broadleaf weed killer and atrazine. Which one will work?

Read the labels carefully before selecting. It seems like you might have a risk of damage to the grass by using an herbicide in very hot temps, but I'm not sure.

Your local extension office would be able to steer you in the right direction.