The only way I know of is modding your Wii to have Bannerbomb to use the Homebrew Channel and get a program that can copy saves to the SD card, and then wait for a mod for Menu 4.3 to put it back onto the new Wii without trouble, at which point you can just uninstall Homebrew Channel as it has served its purpose for moving a valuable save.

Otherwise, if you don't prefer jumping through hoops like I had to to back up my saves in the event of a faulty Wii, then don't bother. Just start a new file on the new Wii and have fun.