Question
What will keep this ground cover?

Answer
I am better at identifying plants than I am at gardening (I have 4 kids and no time), but here's what I've heard:

Bishop's Weed is Aegopodium podagraria, also called Goutweed. I have the variegated form in deep shade in my yard, given to me by my mother about 15 years ago now, and it's hardly bigger than it was then. BUT I have hard, dry, clay soil and a husband who is obsessive with a mower and cuts this stuff down a couple times every year. I've never even seen mine bloom.

From what I understand, though, the green (non-variegated) form is a terrible pest of a weed in gardens, particularly in sun and in loose, sandy soil. It seems to be more common a bit north of here (NJ). Mowing helps some. Certainly cutting any flowers before they go to seed is essential. But for a really bad case you will have to pick by hand (wear gloves, it causes rashes in some people) and then carefully dig the (white) roots out, and then cover with heavy plastic covered with mulch or even with a board. This will, of course, kill other plants with it. Depends on how desperate you are.

Very strong herbicides (Weed B Gone, for instance) should kill it. I don't know that Round-Up would do it. Of course the herbicides would likely kill whatever you have planted near it, and they remain in the soil a long time, not good for kids, pets or new plants.

Simplest answer: give that area up to "ground cover" and start over elsewhere! Just make certain you don't transplant anything from the infested area to any non-infested garden.

Best of luck.