jmc

Let the side length of the cube be $s$. The side length of the cube is equal to diameter of the inscribed sphere, so the radius of the sphere has length $\frac{s}{2}$. Thus, the volume of the sphere is equal to $\frac{4}{3}\pi {\left(\frac{s}{2}\right)}^3=\frac{\pi s^3}{6}$ and the volume of the cube is equal to $s^3$. Hence the ratio of the sphere's volume to the cube's volume is $\boxed{\frac{\pi }{6}}$.