We recently bought a house with a lawn that needs lots of work. The front yard was a disaster of bushes, stumps, weeds, and rocks. We've cleared everything out and gotten to the point where we've cleared out everything we can and planted our first round of grass.

My question is - the one weed I know we haven't really gotten rid of is the soapwort. It's an invasive rhizome type of weed. The previous owner didn't do anything about it and it now occupies about a quarter of the lawn. I've tried digging down and pulling out as much as I can by hand, but it just grows back.

We kept it at bay by mowing it really short, now that I have the lawn planted I want to mow higher.

Has anyone dealt with soapwort in a lawn and what's the magic to controlling it?