On algebraic threefolds whose hyperplane sections are Enriques surfaces

In this paper the following problem is solved: given a singular Fano variety $X$, find a smooth Enriques surface which is an ample Cartier divisor on $X$. The results obtained enable one to construct, using singular Fano varieties, examples of threefolds whose hyperplane sections are Enriques surfaces. They can be used in the classification of log-Fano varieties of (Fano) index 1.