We aim at getting high spatial resolution information on the dusty core of bipolar planetary nebulae to directly constrain the shaping process. Methods: We present observations of the dusty core of the extreme bipolar planetary nebula Menzel 3 (Mz 3, Hen 2-154, the Ant) taken with the mid-infrared interferometer MIDI/VLTI and the adaptive optics NACO/VLT. The core of Mz 3 is clearly resolved with MIDI in the interferometric mode, whereas it is unresolved from the Ks to the N bands with single dish 8.2 m observations on a scale ranging from 60 to 250 mas. A striking dependence of the dust core size with the PA angle of the baselines is observed, that is highly suggestive of an edge-on disk whose major axis is perpendicular to the axis of the bipolar lobes. The MIDI spectrum and the visibilities of Mz 3 exhibit a clear signature of amorphous silicate, in contrast to the signatures of crystalline silicates detected in binary post-AGB systems, suggesting that the disk might be relatively young. We used radiative-transfer Monte Carlo simulations of a passive disk to constrain its geometrical and physical parameters. Its inclination (74 degrees $pm$ 3 degrees) and position angle (5 degrees $pm$ 5 degrees) are in accordance with the values derived from the study of the lobes. The inner radius is 9$pm$ 1 AU and the disk is relatively flat. The dust mass stored in the disk, estimated as 1 x 10-5Msun, represents only a small fraction of the dust mass found in the lobes and might be a kind of relic of an essentially polar ejection process.
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