Greens Bottom is a trailhead in the southern part of St Charles, so it is in an eastern direction along the trail from Weldon Spring (northeast actually). It does not appear on the maps because it is just a trailhead, rather than a town. But here's a page with
info on Greens Bottom including a map and directions.
From Weldon Spring to Greens Bottom is about 10 miles each way. There are heavy woods on either side of the trail, making this section a favorite for many, and a few nice views of the Missouri River too. There are no businesses between (or at) Weldon Spring and Greens Bottom.