  It was proved by J. A. Chen and M. Chen that a terminal Fano $3$-fold $X$
satisfies $(-K_X)^3\geq \frac{1}{330}$. We show that a non-rational
$\mathbb{Q}$-factorial terminal Fano $3$-fold $X$ with $\rho(X)=1$ and
$(-K_X)^3=\frac{1}{330}$ is a weighted hypersurface of degree $66$ in
$\mathbb{P}(1,5,6,22,33)$.
