A macroscopically oriented inverse hexagonal phase (H-II) of the lipid phytantriol in water is converted to an oriented inverse double diamond bicontinuous cubic phase (Q(II)(D)). The initial H-II phase is uniaxially oriented about the long axis of a capillary with the cylinders parallel to the capillary axis. The H-II phase is converted by cooling to a Q(II)(D) phase which is also highly oriented, where the cylindrical axis of the former phase has been converted to a < 110 > axis in the latter, as demonstrated by small-angle X-ray scattering. This epitaxial relationship allows us to discriminate between two competing proposed geometric pathways to convert H-II to Q(II)(D). Our findings also suggest a new route to highly oriented cubic phase coatings, with applications as nanomaterial templates.