Unofficially, you need at least 2GB. Most of the benefits from 64-bit computing come when you have the extra RAM. Remember, 32-bit Windows will only see about 3.25GB of RAM (the rest is secretly used behind the scenes).
Also, many developers tend to expect more system resources for 64-bit operating systems (i.e. som eSystem Requirement pages say that you need 1GB of RAM, but you need 2GB of RAM for 64-bit). I think this is because games are still 32bit apps, and have to be emulated on Windows 64-bit. More info on the emulator here.
If you're concerned about having a dual-core CPU, don't worry, as 32-bit Windows will see and use both cores. The main difference is that 32bit Windows will put ~ 50% of programs on one core, and 50% on the other. If you run Windows 64-bit most of your programs (and games) are still 32bit, so you don't get much benefit, as it works in the same way. The biggest benefit comes when you run 64-bit programs on 64-bit Windows, as the program can run on more than one core at the same time. Video editing programs, for example, are much better if they are 64-bit (even better if they support nVidia CUDA for making use of the GPU as well).
/technical lesson