As shown in Fig. 6.1, let the centroids of faces $ABC$ and $A_1B_1C_1$ be $O$ and $O_1$, respectively. Denote the midpoint of segment $OO_1$ as $P$. By symmetry, we know that $P$ is the centre of the circumscribed sphere of the regular triangular prism and $PA$ is its radius.

Fig. 6.1

It is easy to find that $PO \perp AO$, and thus
$$PA = \sqrt{PO^2 + AO^2} = \sqrt{\left(\frac{3}{2}\right)^2 + (\sqrt{3})^2} = \frac{\sqrt{21}}{2}.$$
Therefore, the volume of the circumscribed sphere is $\frac{4}{3}\pi\left(\frac{\sqrt{21}}{2}\right)^3 = \frac{7\sqrt{21}}{2}\pi$.