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

On $(2,4)$ complete intersection threefolds that contain an Enriques surface