After taking what was probably the best photo of a web I'll ever get, I decided to keep taking pics and do a little research on the spinner who had built it.
She's a tiny and really odd-looking critter, resembling a crab more than a spider, with a bright yellow and black shell with red spines sticking out of it. Since she was the size of a jujube and hanging in mid-air, which makes it almost impossible to focus, this is the best close-up shot I could get of her:
She caught a lot of bugs every day, and also constantly repaired her web and kept it very clean (here you can see where she patched up the big hole in her web that was in the original shot):
We tried not to disturb her for as long as possible, but she was anchored to a high-traffic area of the lawn, and we had to mow the grass, so we took down her web and relocated her to a safer spot in the yard under the big oak tree.
Two days later, she had rebuilt the entire web:
After looking around on the internet I believe she's a species of Gasterocantha cancriformis, Spiny Orb Weaver. This photo on Flickr shows much better detail than mine, and is almost the same, except my spinner is yellow and black and has more red spines.