  We study the X(3872)-->D^0 \bar D^0 \pi^0 decay within a D \bar D^* molecular
picture for the X(3872) state. This decay mode is more sensitive to the
long-distance structure of the X(3872) resonance than its J/\psi\pi\pi and
J/\psi3\pi decays, which are mainly controlled by the details of the X(3872)
wave function at short distances. We show that the D^0 \bar D^0 final state
interaction can be important, and that a precise measurement of this partial
decay width can provide valuable information on the interaction strength
between the D^{(*)} \bar D^{(*)} charm mesons.
