I simply adore Google Maps and Google Earth.
If you download Google Earth to your computer, and wait for the images, etc. to load, you can have a lot of fun. The downtown St. Louis buildings feature is pretty cool, although it's kind of annoying they depict the Arch as if it was just a monolithic gray box sticking up into the sky. But, hey, for free, how advanced do you want your 3-D renderings?
Among the many layers you can turn on are Crime Stats and Census profiles. While the Crime Stats are pretty much useless, and the Census data is very basic, these data sets are aggregated by county, so they indicate the approximate center point of each county.
So, from either of those points, we can estimate the location of the geographic center of any given county - including the City of St. Louis.
Geographic center point for selected counties
Those last four are all pretty much rural locations, but near their respective county seats. Interesting that two of them are on or near camp grounds.
Source: Google Earth, 10/11/05.