Actually, there is. If you haven't noticed already, the connection port for your Gamecube is identical to that of the N64; people have already used their N64 Svideo cable on their GC without any problems. Search eBay for a N64 or Gamecube VGA box, you'll find some. They go for around $50.
It sounds like a very inexpensive line doubler device, like the PS2 VGA adaptor. It would have to be in order to work with games that do not output at a 480p scanning frequency.