We present first observations of Mira stars obtained at the Very Large Telescope Interferometer (VLTI) at 2 microns, equipped with the VINCI instrument, using different baselines with both the test siderostats and the 8.2 m unit telescopes. These observations, collected in the course of the VLTI commissining program, have targeted so far about sixty cool giant stars. In this talk, we list and describe the measurements obtained for 14 Mira stars, many of them representing first-time determinations of the visibility. In particular, we devote special attention to a study of Mira itself, for which very accurate determinations of the visibility at several baseline lenghts and orientations could be obtained. We find that a two-component model is more consistent with the data than a single stellar disc. Further observations are needed for a better understanding of this source. The VLTI will constitute an ideal instrument for such studies in the future. In particular it will permit detailed investigations of southern AGB stars, such as accurate measurements of surface structureparameters (diameters, diameter variations, asymmetries, center-to-limbvariations, special features like hot spots) and of circumstellar envelopes.PUBLICATION ABSTRACT