Of course you can use that disc. you bought it, its yours. You can use that disc over and over again, and it also has some nice features for repairing a computer. Anyways yes you can use the 64 bit disc in a clean install, just make sure you used the same license key you used on the 32 bit version. Also the both have to be the same product type, for example:
home premium 32bit= home premium 64 bit
ultimate 32 bit = ultimate 64 bit
You can't go from your home premium 32 bit = your new ultimate 64 bit.
Must be the original product type, and product key. Just insert disc, the same key and clean install. Good luck
And there shouldn't be any problems, unfortunately you will lose everything, so backup important data.
If problems do arise, in which they shouldn't then feel free to post back on seven forums.
Edit: this may help : Clean Install with a Upgrade Windows 7 Version *also if you find a windows.old folder on your new version, and it contains files you don't want Then open disk cleanup and you must chose system files select all, and this will rid of it.*