According to authorized GameCenter page at Apple, there’s no GameCenter for iPhone 3G in forthcoming iOS 4.1 keep informed.

Apple iOS 4.1 is coming this week (Sep 8). Surrounded by the main updates so as to it brings is the GameCenter. According to Apple:
Game Center lets friends — and soon-to-be-friends — in on the act. Request an important person to join, and then get a game going. Or go up alongside people you don’t know, from wherever in the world, in a multiplayer game. It’s your world. Everybody moreover is just playing in it.
Apple has supported subsequent devices to get GameCenter in iOS 4.1:
iPhone 4
iPhone 3GS
iPod Touch 4G
iPod Touch 3G
iPod Touch 2G