  Let $V$ be a smooth projective 3-fold of general type. Denote by $K^{3}$, a
rational number, the self-intersection of the canonical sheaf of any minimal
model of $V$. One defines $K^{3}$ as a canonical volume of $V$. The paper is
devoted to proving the sharp lower bound $K^{3}\ge {1/420}$ which can be
reached by an example: $X_{46}\subseteq \mathbb{P}(4,5,6,7,23)$.
