As a rule of thumb, I wouldn't go one byte less than 512MB for 32-bit, or 1GB for 64-bit.
More realistically, 1GB is a probably a good "minimum RAM" to start with for any GUI OS (32- or 64-bit), then increase as-needed. People might disagree, but it's honestly my experience that if you really need to have less than 1GB RAM ... then you probably shouldn't be considering a Gnome or KDE desktop. If you don't have enough RAM, you should probably consider a server (e.g. text-mode) environment, or a distro like Puppy Linux (which supports REALLY tiny memory footprints!), or Ubuntu Remix.