Does the new Gamecube controller work with Gamecube games on a Wii

According to the specifications found on Nintendo's E3 website, the drive doesn't accept GameCube's small 8cm discs, only the new high-density discs for Wii U and the normal 12cm Wii discs - so no, it won't be compatible.

This also fits with the likely strategy they use for backwards compatibility; they shut all the new stuff off so the console behaves just like the Wii - that's how they do it on the Wii for GC stuff, on the 3DS for DS stuff, and on the DS for GBA stuff. AFAIK, that technique doesn't work very well if you're trying to chain it like this, since the Wii part would have to turn all of its stuff off to switch to GameCube mode, and that ends up getting way too messy.

As @Powerlord pointed out, the GameCube Super Smash Bros. Edition Controller is compatible with the original GameCube and Wii. This is confirmed on Nintendo's website (in the same link provided, very last question)

Can the GameCube Super Smash Bros. Edition Controller be used with a GameCube or a compatible Wii model?

Yes, it can be used with these systems.

Sidebar: Always make sure to check the box before confirming your purchase! 3rd-party companies like Hori and PDP make their own "Fight pads" as well, but they're Classic Wii controllers without the analog triggers. (They have squishy triggers, but there's no analog signal, and that means tossing your shield or grab is delayed significantly.