Is the Wii U compatible with Gamecube games?

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.

Here's a short article with confirmation from a Nintendo source that the Wii U will not be able to play Gamecube games.

From what information has been released so far, the system does not accept the 8cm GameCube discs, nor does it have ports for the controllers/memory cards.

However, this doesn't rule out alternate means of playing GameCube games on the WiiU, such as with direct downloads (a la Wii's Virtual Console).

While not official, Nintendon't is WiiU homebrew that will allow for the playing of Gamecube ISO's, so you could use your Wii to rip your own discs and play them on the WiiU. This homebrew even supports the gamecube adapter.