Although early reports would have you believe the answer is five (Nature1985, 318, 162)*, these videos could muddy the waters somewhat.
They appear (along with several other videos) on a website from New York City's Whitney Museum, which just this weekend opened a new exhibit on R. Buckminster Fuller, the geometric visionary/architect who is C60's namesake.
Watch closely: Is the video of museum employees putting together a geodesic dome a stop-motion video (like claymation) or is it merely sped up? Also, that origami buckyball looks mind-bogglingly hard to fold. (It took me days to master paper-crane folding in second grade.) If there are any detailed instructions for how to fold it, though, we'd love to hear about it!
*What I love most about this classic paper is the elegant simplicity of Figure 1.