1. **Problem:** Find the largest volume of a cube that can be enclosed in a sphere of diameter 2 cm. 2. **Understanding the problem:** The cube is inscribed inside the sphere, so the sphere touches all vertices of the cube. 3. **Formula:** The diagonal of the cube equals the diameter of the sphere. 4. Let the edge length of the cube be $a$. The space diagonal of a cube is given by: $$\text{diagonal} = a\sqrt{3}$$ 5. Since the cube is inscribed in the sphere, the diagonal equals the diameter of the sphere: $$a\sqrt{3} = 2$$ 6. Solve for $a$: $$a = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$$ 7. Volume of the cube is: $$V = a^3 = \left(\frac{2\sqrt{3}}{3}\right)^3 = \frac{8 \times 3\sqrt{3}}{27} = \frac{8\sqrt{3}}{9}$$ 8. Simplify the volume: $$\frac{8\sqrt{3}}{9} = \frac{8}{3\sqrt{3}}$$ 9. **Answer:** The largest volume of the cube is $\boxed{\frac{8}{3\sqrt{3}}}$ cm³. This corresponds to option (4).