  Bipolarity in proto-planetary and planetary nebulae is associated with events
occurring in or around their cores. Past infrared observations have revealed
the presence of dusty structures around the cores, many in the form of disks.
Characterising those dusty disks provides invaluable constraints on the
physical processes that govern the final mass expulsion of intermediate-mass
stars. We focus this study on the famous M2-9 bipolar nebula, where the moving
lighthouse beam pattern indicates the presence of a wide binary. The compact
and dense dusty core in the center of the nebula can be studied by means of
optical interferometry. M2-9 was observed with VLTI/MIDI at 39-47 m baselines
with the UT2-UT3 and UT3-UT4 baseline configurations. These observations are
interpreted using a dust radiative transfer Monte Carlo code. A disk-like
structure is detected perpendicular to the lobes and a good fit is found with a
stratified disk model composed of amorphous silicates. The disk is compact,
25$\times$35 mas at 8$\rm \mu m$, and 37$\times$46 mas at 13$\rm \mu m$. For
the adopted distance of 1.2 kpc, the inner rim of the disk is $\sim$15 AU. The
mass represents a few percent of the mass found in the lobes. The compactness
of the disk puts strong constraints on the binary content of the system, given
an estimated orbital period 90-120yr. We derive masses of the binary components
between 0.6--1.0M$_{\sun}$ for a white dwarf and 0.6--1.4M$_{\sun}$ for an
evolved star. We present different scenarios on the geometric structure of the
disk accounting for the interactions of the binary system, which includes an
accretion disk as well.
