I'm gonna guess it is a registry error due to multiple versions of Civ III.
I would un-install all versions of the game, making sure you've made copies of all the saves.
I would then use Windows Explorer to manually delete any Civ III-ish folders that remain. This can be tricky, since the different owners of Civ III used different directories. So check Program Files/Atari and Program Files/Infogrames (?) to be sure. Don't trust the un-install to find/remove/fix everything.
Next, I would download WinASO Registry Optimizer, a freeware utility, and run it several times. The freeware version will only fix 10 errors at a time, but you can run it as many times as you need. This will remove any traces of Civ III in the registry that point to phantom locations on the hard drive. And fix other things, too.
(You could also defrag the hard drive at this point. It won't fix anything but should make everything work a bit faster. And perhaps download/run SpyBot Search and Destroy, Spywareblaster and AdAware. These three things will look for and remove spyware and such. All are free downloads. I mean, if you're going to take the time to do some PC housecleaning, do it well!

)
Now I would reinstall Civ III Complete. It will not need to be patched, since this version has all the patches already installed.
At this point, you should have a clean install and the game should play just fine.