A two-dimensional dilatonic black hole induced by a topological soliton is exactly solvable in the scalar field theory coupled to dilaton gravity. The Hawking radiation of the black hole is studied in the one-loop approximation with the help of the trace anomaly of energy-momentum tensors which is a geometrical invariant. The quantum theory can also be solvable in the RST scheme in order to consider the back reaction of the metric. The energy of the black-hole system is calculated and the classical thunderpop energy corresponding to the soliton energy is needed to describe the final state of the black hole. Finally we discuss the possibility of conservation of the topological charge.