My Mac Mini won't acknowledge Snow Leopard disk to upgrade OS. DVD drive works fine.

Purchased my Mini January 2009 with 2 GHz Intel Core 2 Duo processor w/ 2 GB 667 MHz DDR2 SDRAM. Trying a long-overdue upgrade from OS 10.5.8 to Snow Leopard, the computer won't acknowledge the disk, and the DVD drive runs other disks.

According to other links here, I should be able to upgrade through OS 10.7 without problem, correct? (FYI, I have four USB ports on this thing.) Also, I noticed my Snow Leopard disk - purchased from Apple, not bootleg - is 10.6.3, which is what they shipped when I said OS 10.5.8 is in here. Am I missing an earlier version than 10.6.3? In any case, the disk just grinds awhile, then spits out. Nothing ever shows on the display.

I have 59 gigs of internal hard drive space available.

I'm not really tech savvy, so any thoughts?