The potentialities of an argon Dark Matter detector with real-time
directional information located at Laboratori Nazionali del Gran Sasso (LNGS) are discussed. Columnar recombination combined with a dual-phase Liquid Argon Time Projection Chamber (TPC) provides a promising approach for a directional detector. Even in the case of a minimal angular resolution, a very clear signature of WIMP scattering on target nuclei is shown to be expected. Indeed, the ratio of horizontal to vertical events changes by a factor of four during the day and has a characteristic sidereal-day periodicity. Demonstration of a directional sensitivity in a large Liquid Argon TPC could have high impact on the future experiments in the field, opening up the potential to find conclusive evidence for dark matter or disprove the WIMP hypothesis at and above the mass range explored by planned accelerator experiments.