Non-invasive and "moderate to fast ground cover" are competing terms. Is this full sun (sounds like it)? If so, you might look up any of a number of sedums. S. kamtschaticum or S. kamtschaticum ellacombianum are "clumping" forms that don't run but the clump expands slowly. Be aware that a common mistake with sedum ground covers is to use mulch - many varieties expand much more rapidly on bare soil. Sedum with pre-emergent weed control can get them going, but read the label on the pre-emergent to make sure that it is compatible with your type of sedum.
Another fairly controllable type of g.c. for sun are the low junipers, J. procumbens and J. horizontalis varieties. Gotta watch them for mites, however. Mulch does work with them.
Good luck!

I've been really frustrated with weeding too, living on 2 acres with lots of grass to invade beds and lots of flower beds I kept up with until I was gone for the whole month of June one year and all the grasses went to seed. I spend a lot of time I don't want weeding, but I've seen progress in a few beds that actually don't need weeding this year. It's largely due to groundcovers but sometimes to just having the perennials close enough to each other to really fill in. I like Pulmonaria for that, they have lovely spring bloom and then nice variegated foliage and self-sow, but are low and complementary to other perennials.

As Hortster points out, gc that really fill in are by nature invasive. I use some invasive things that still don't overrun perennials.

Herb Robert is one that helps, probably regarded with horror by most, but it is very easy to pull out and really covers ground.

Then I have a little wild violet with heart-shaped leaves that spreads fast and is very low.

I also use some variegated Vinca Minor, Acaena saccaticulpula "Blue Haze", Acaena inermis purpureum, Ajuga, and in the fall I sow corn salad in beds, at least here it grows all winter and is edible, then makes tiny white flowers in spring and self-sows, then disappears. It helps occupy ground so the weeds can't in early spring.

Another wonderful plant in that respect is Anemone nemarosa, a rhizome that makes a nice clump 3" tall, and is covered with flowers in early spring, shutting out weeds, then goes dormant by summer and disappears. I'm marking where I have nice clumps and digging in late summer and spreading them through my beds.

I'm spreading a couple of other gc's in one bed as an experiment, Lamium maculatum "beacon silver", and the tiny wild lawn daisy Bellis ? that blooms a lot, clumps up, and self-sows.

I have Columbine sown from seeds (and they self-sow) that sometimes I regard as a weed, especially when I have to trim off all the spent flower stalks, but I have a dwarf form that stays low. They are blooming now and make all the beds look like fairy-land. I'm planning to collect all the seeds and spread them all over one very weedy bed that has come up solid grass, in hopes they will shade it out over a few years. The foliage is nice, a blue green low clump. Anyways, that's my current strategy.

Nancy

Just saw this post and thought I would throw in my 2cents in case you are still looking.

Last year I planted a flat of Ajuga Chocolate Chip, all around my front and back yards, in full sun. None of these were planted closer that 18" from each other. It grows very fast. So fast that there is NO bare earth between any of them any more and I've started pulling out huge chunks and moving them else where. I also like the fact that they are low to the ground so all my perinneals and shrubs just rise above them. It's also exremely dense (at least in my garden) the weeds don't stand a chance. Most of my beds are raised and the ajuga holds the soil in position very well. When I put in these beds I used an edger to put in a V trench. I put the ajuga in the trench as an edger and just mow over it. I still find a stray piece of grass growing up through it but I just pull it out and those stray pieces are becoming less and less.

Alexis