  We study the thermal physics of AdS(3) and the BTZ black hole when embedded
in String theory. The exact calculation of the Hagedorn temperature in TAdS(3)
is reinterpreted as the appearance of a winding tachyon both in AdS(3) and BTZ.
We construct a dual framework for analyzing the phases of the system. In this
dual framework, tachyon condensation and geometric capping appear on the same
footing, bridging the usual gap of connecting tachyon condensation to
modifications of geometry. This allows us to construct in a natural way a
candidate for the unstable phase, analogous to a small black hole in higher
dimensions. Additional peculiar effects associated with the Hagedorn
temperature and the Hawking-Page transition, some to do with the asymptotic
structure of AdS(3) and some with strong curvature effects, are analyzed and
explained.
