  We consider Minkowskian Jackiw-Teitelboim (JT) gravity in Bondi gauge at
finite temperature, with non-zero vacuum energy. Its asymptotic symmetries span
an extension of the warped Virasoro group, dubbed "BMS$_2$", which we
investigate in detail. In particular, we show that this extension has a single
coadjoint orbit when central charges are real and non-zero. The ensuing
BMS-Schwarzian action has no saddle points, and only coincides with the
boundary action functional of flat JT gravity up to a crucial dilatonic
zero-mode that ensures the existence of a well-defined bulk variational
principle. We evaluate the corresponding gravitational partition function,
which turns out to be one-loop exact precisely thanks to the presence of such a
zero-mode.
