I suspect its the FX 5200 card.
That card was rushed out to replace the disasterous MX series of chips. The FX 5200 ATI equivalent was the 9000 & 9200 series of cards - its a low end low performing card. The FX had a troubled time getting going in the marketplace. Despite its numerical designation (5200) it was worse than the MX 440s on release... they improved it in the end, particularly the driver.
Your drivers are uptodate which is good
As always someone on Planet Earth will be running Civ on it, but you will have to work at it. You need to reduce as much as possible the load on the video ram by reducing the resolution, low textures, stopping unnecessary background programs via MSconfig etc - anything that will reduce the load on the card. There are numerous ways to reduce the load depending on your setup, lots of posts go thro this on the board, have a look at the first two pages. The inherent weakness of the card and the fact that its only got 64Mb of Video RAM is your issue.
I would seriously think about replacing the Video Card, you would notice a significant improvement. Research your motherboard carefully tosee what card it could take, being careful over heating issues
Regards
Zy