$2 \pi / 3$. The solid consists of two cones, one whose base is the circle $x^{2}+y^{2}=1$ in the $x y$-plane and whose vertex is $(0,0,1)$, and the other with the same base but vertex $(0,0,-1)$. Each cone has a base area of $\pi$ and a height of 1, for a volume of $\pi / 3$, so the answer is $2 \pi / 3$.