Sadly, there have been a few points in the past when Debian (and Ubuntu/Mint, by extension) have dropped drivers for older graphics cards from their standard installs. Luckily, Debian makes it easy to install using a safe, generic driver, and then recompile the kernel and install proprietary graphics drivers.
I tried installing Mint and Ubuntu on a laptop with similar specs; the live CD worked fine but the screen went black after I installed to hard drive. I got OpenSUSE and Fedora to work okay, but it was too much of a pain to install the proprietary graphics drivers.
Eventually I settled on installing from a Debian Live CD. Not only was my system usable after first install, but I was able to return later and recompile the kernel with the older NVIDIA driver. Thankfully the Debian repos make this process extremely easy! On 1 GB RAM I'm running the Xfce desktop just fine.
I love Linux Mint, but I would definitely recommend going with pure Debian on really old PC's. Saline OS (with Xfce desktop) is a good Debian Live CD for systems built during the Pentium 4 days. Systems older than P4 can still be useful with CrunchBang--OpenBox is even less memory-intensive than Xfce!