Short answer: you can't. The Nintendo DS and the GameCube are fundamentally incompatible, and no adapter, hack, or official method bridges that gap. Here's why—and what you can do.
The GameCube's IBM PowerPC "Gekko" CPU and ATI "Flipper" GPU are vastly more powerful than the DS's dual ARM processors (ARM946E-S at 67 MHz and ARM7TDMI at 33 MHz). The DS simply lacks the raw horsepower to emulate or natively run GameCube code.
GameCube games come on 8 cm proprietary optical discs; the DS uses game cards (cartridges) and, later, DSiWare downloads. Each console runs a proprietary OS tailored to its specific hardware—there's no cross-compatibility layer.
There is no safe, legal way to run GameCube software directly on DS hardware. Focus on the DS's own library or emulate responsibly on a PC.