6. As shown in Figure 1, in the unit cube $A B C D$ $A_{1} B_{1} C_{1} D_{1}$, $O$ is the center of the square $A B C D$, points $M$ and $N$ are on the edges $A_{1} D_{1}$ and $C C_{1}$ respectively, with $A_{1} M=$ $\frac{1}{2}$ and $C N=\frac{2}{3}$. Then the volume $V$ of the tetrahedron $O M N B_{1}$ is
6. $\frac{11}{72}$.

Taking $A$ as the origin, $\overrightarrow{A B} 、 \overrightarrow{A D} 、 \overrightarrow{A A_{1}}$ as the $x$-axis, $y$-axis, and $z$-axis respectively to establish a rectangular coordinate system. Then
$$
\begin{array}{l}
O\left(\frac{1}{2}, \frac{1}{2}, 0\right), M\left(0, \frac{1}{2}, 1\right), \\
N\left(1,1, \frac{2}{3}\right), B_{1}(1,0,1) . \\
\text { Hence } V=\frac{1}{6}\left|\overrightarrow{O B_{1}} \times \overrightarrow{O N} \cdot \overrightarrow{O M}\right| \\
=\frac{1}{6}\left|\left(\frac{1}{2},-\frac{1}{2}, 1\right) \times\left(\frac{1}{2}, \frac{1}{2}, \frac{2}{3}\right) \cdot\left(-\frac{1}{2}, 0,1\right)\right| \\
=\frac{11}{72} .
\end{array}
$$