The recommended way to install **any** proprietary (non-open-source) hardware drivers is to go to Menu > Admin > Hardware Drivers, and click Activate if available. This means don't go to their website, download their linux drivers and ask now what. It also means don't use EnvyNG, as I have yet to hear of a success story from this method. If you know what you're doing, you could use these methods, and they may well work for you, but don't expect much in the way of support if something goes wrong.
If there is no Activate option, with your card is not listed in Hardware Drivers, then there are no proprietary drivers that support both the card and the installed system, so you have to use the open-source drivers. Your card may well be supported on an older version of Mint though, e.g. the LTS release, so if the open-source driver is not good enough for you, then installing the older version of Mint is your only option. Don't try to force the drivers to install, from either the website, or EnvyNG, it won't work, and you'll just have to recover from non-graphical mode. The reason for this is that the driver is not written to take advantage of the latest technology available in the newer releases.
ATI dropped support for a lot of cards back in Feb '09, even on Windows, see http://support.amd.com/us/gpudownload/l ... linux.aspx for a list, where you'll see they list the older driver v9.3 Again, because newer versions of Mint, or any Linux released since that date use newer technology that won't work with the driver, don't try to install it. Try out the open-source driver for a while, you might find it's good enough. If it isn't, your only option is to install Felicia and wait until the open-source driver gets better.
As an explanation as to why the proprietary driver is so much better than the open-source one, consider that ATI's driver writers have full access to all the hardware specs and strangeness of the cards themselves. In contrast the open-source dudes only have the physical cards themselves to work with, so they're trying to reverse-engineer all the work ATI will have already done on documenting the card. The good news is that when ATI drops support, they tend to release more hardware specs, meaning that the open-source dudes can have a massive boost in productivity, so you can expect the open-source drivers support to rapidly improve in future releases. As such, you should keep trying out newer versions of Mint as they are released, to see it it's good enough for you yet (or until the new features convince you to put up with minor glitches

EDIT: there's reports that the karmic open-source drivers work well on at least the X1200, so if you're caught by this problem, please try Helena when it comes out.