  We study nodal complete intersection threefolds of type $(2,4)$ in $\PP^5$
which contain an Enriques surface in its Fano embedding. We completely
determine Calabi-Yau birational models of a generic such threefold. These
models have Hodge numbers $(h^{11},h^{12})=(2,32)$. We also describe Calabi-Yau
varieties with Hodge numbers equal to $(2,26)$, $(23,5)$ and $(31,1)$. The last
two pairs of Hodge numbers are, to the best of our knowledge, new.
