Del Pezzo surfaces with nonrational singularities

Normal algebraic surfaces $X$ with the property $\operatorname{rk}(\operatorname{Div}(X)\otimes\mathbb Q/{\equiv})=1$, numerically ample canonical classes, and nonrational singularities are classified. It is proved, in particular, that any such surface $X$ is a contraction of an exceptional section of a (possibly singular) relatively minimal ruled surface $\widetilde X$ with a nonrational base. Moreover, $\widetilde X$ is uniquely determined by the surface $X$.