3. The volume of the circumsphere of the smaller tetrahedron formed by the centers of the four faces of a regular tetrahedron with edge length 1 is $\qquad$
3. $\frac{\sqrt{6} \pi}{216}$.

Let $O_{1}$ and $O_{2}$ be the centers of the faces $ABC$ and $ACD$ of the regular tetrahedron $ABCD$, respectively, and let $AO_{1}$ and $AO_{2}$ intersect the edges $BC$ and $CD$ at points $E$ and $F$, respectively. Then, $EF$ is the midline of $\triangle BCD$.
Thus, $EF=\frac{1}{2}$.
Therefore, $O_{1}O_{2}=\frac{2}{3} EF=\frac{1}{3}$, which means the edge length of the smaller regular tetrahedron is $\frac{1}{3}$.

Hence, the edge length of its circumscribed cube is $\frac{\sqrt{2}}{6}$, and the radius of its circumscribed sphere is $\frac{\sqrt{6}}{12}$, with a volume of $\frac{\sqrt{6} \pi}{216}$.