  We prove that no local diffeomorphism invariant two-dimensional theory of the
metric and the dilaton without higher derivatives can describe the exact string
black hole solution found a decade ago by Dijkgraaf, Verlinde and Verlinde. One
of the key points in this proof is the concept of dilaton-shift invariance. We
present and solve (classically) all dilaton-shift invariant theories of
two-dimensional dilaton gravity. Two such models, resembling the exact string
black hole and generalizing the CGHS model, are discussed explicitly.
