jmc

Let $A$ be the point on the hemisphere where the top of the hemisphere touches the sphere, and let $B$ be the point on the hemisphere where the base of the hemisphere touches the sphere. $AB$ is a diameter of the sphere and a radius of the hemisphere. Thus, the diameter of the sphere is 2, so the radius of the sphere is 1 and the volume of the sphere is $\frac{4}{3}\pi (1^3)=\boxed{\frac{4}{3}\pi }$.