  The supergravity dual of $N$ regular and $M$ fractional D1-branes on the cone
over the Einstein manifold $Q^{1,1,1}$ has a naked singularity in the infrared.
The supergravity dual of $N$ regular and $M$ fractional D3-branes on the
conifold also has such a singularity. Buchel suggested and Gubser et al. have
shown that in the D3-brane case, the naked singularity is cloaked by a horizon
at a sufficiently high temperature. In this paper we derive the system of
second-order differential equations necessary to find such a solution for
$Q^{1,1,1}$. We also find solutions to this system in perturbation theory that
is valid when the Hawking temperature of the horizon is very high.
