Mori Structures on a Fano Threefold of Index 2 and Degree 1

It is proved that the Mori structures on a nonsingular Fano threefold of index 2 and degree 1 are represented precisely by this Fano variety itself and by fibrations into del Pezzo surfaces of degree 1 that emerge from the blowups of curves of arithmetic genus 1 and degree 1. In particular, such a Fano variety is nonrational and all its birational automorphisms are regular.