jmc

We draw a diagram:



The sphere's diameter length is equal to the big cube's side length, which is 9.



Now the sphere's diameter is equal to the space diagonal of the small cube, meaning that the distance between two opposite corners of a cube is equal to the diameter of the sphere. To compute the space diagonal of the cube, let the side length of the cube be $s$, and label points $A$, $B$, $C$, $D$, $E$ as shown below.

We look at triangle $△BDE$, where $\overline{BE}$ is the space diagonal. $\overline{DE}$ is a side length of the cube with length $s$. $\overline{BD}$ is the hypotenuse of an isosceles right triangle with legs length $s$, so its length is $\sqrt{s^2+s^2}=s\sqrt{2}$. So we have $$BE=\sqrt{D{E}^2+B{D}^2}=\sqrt{s^2+(s\sqrt{2}{)}^2}=\sqrt{3s^2}=s\sqrt{3}.$$Thus, the space diagonal of a cube with side length $s$ has length $s\sqrt{3}$. The sphere has diameter 9, which is equal to the space diagonal of the cube, so we have $$9=s\sqrt{3}\Rightarrow s=\frac{9}{\sqrt{3}}.$$Finally, the volume of the cube is $s^3={\left(\frac{9}{\sqrt{3}}\right)}^3=\boxed{81\sqrt{3}}$.