  We consider the Hamiltonian dynamics and thermodynamics of the
two-dimensional vacuum dilatonic black hole in the presence of a timelike
boundary with a fixed value of the dilaton field. A~canonical transformation,
previously developed by Varadarajan and Lau, allows a reduction of the
classical dynamics into an unconstrained Hamiltonian system with one canonical
pair of degrees of freedom. The reduced theory is quantized, and a partition
function of a canonical ensemble is obtained as the trace of the analytically
continued time evolution operator. The partition function exists for any values
of the dilaton field and the temperature at the boundary, and the heat capacity
is always positive. For temperatures higher than $\beta_c^{-1} =
\hbar\lambda/(2\pi)$, the partition function is dominated by a classical black
hole solution, and the dominant contribution to the entropy is the
two-dimensional Bekenstein-Hawking entropy. For temperatures lower
than~$\beta_c^{-1}$, the partition function remains well-behaved and the heat
capacity is positive in the asymptotically flat space limit, in contrast to the
corresponding limit in four-dimensional spherically symmetric Einstein gravity;
however, in this limit, the partition function is not dominated by a classical
black hole solution.
