  We construct the first known example of a near horizon supergravity solution
for a pair of NS5-branes, intersecting on R^{1,3} and localised in all
directions except a single transverse circle. We do this by establishing an
explicit map between the conifold metric and the near horizon geometry of two
intersecting NS5-branes, clarifying and correcting a number of open issues in
the literature en route. Our technique is general in nature and may be applied
to a whole class of 1/4-BPS five-brane webs and their geometric duals. These
1/4-BPS solutions may have an interesting holographic interpretation in terms
of little string theory.
