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We prove that no local diffeomorphism invariant two-dimensional theory of themetric and the dilaton without higher derivatives can describe the exact stringblack hole solution found a decade ago by Dijkgraaf, Verlinde and Verlinde. Oneof the key points in this proof is the concept of dilaton-shift invariance. Wepresent and solve (classically) all dilaton-shift invariant theories oftwo-dimensional dilaton gravity. Two such models, resembling the exact stringblack hole and generalizing the CGHS model, are discussed explicitly.Comment: 24 pages, 3 eps-figures, revised version (more references, clarified  some of the discussion

Non-existence of a dilaton gravity action for the exact string black hole