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

The sharp lower bound for the volume of 3-folds of general type with \chi(\Co{X})=1