We study the duality between string theory formulated on a curved exact background (the two-dimensional black hole) and string theory in flat space with a tachyon-like potential. We generalize previous results on this subject by discussing a twisted version of the Fateev-Zamolodchikov-Zamolodchikov conjecture (FZZ). This duality is shown to hold at the level of N-point correlation functions on the sphere topology, and connects tree-level string amplitudes in the Euclidean version of the 2D black hole to correlation functions in a nonlinear sigma-model in flat space but in presence of a tachyon wall potential and a linear dilaton. The dual CFT we propose here corresponds to the perturbed 2D quantum gravity coupled to c < 1 matter, where the operator that describes the tachyon-like potential can be seen as an n = 2 momentum mode perturbation, while the usual sine-Liouville potential involved in the FZZ duality would correspond to the vortex sector n = 1. We give a precise prescription for computing correlation functions in the twisted model.