Context. In 1996, Sakurai's object (V4334 Sgr) suddenly brightened in the center of a faint Planetary Nebula (PN). This very rare event was interpreted as being the reignition of a hot white dwarf that caused a rapid evolution back to the cool giant phase. From 1998 on, a copious amount of dust has formed continuously, screening out the star that remained embedded in this expanding high optical-depth envelope. Aims. We present observations that we use to study the morphology of the circumstellar dust to investigate the hypothesis that Sakurai's Object is surrounded by a thick spherical envelope of dust. Methods. We acquired unprecedented, high angular-resolution, spectro-interferometric observations, with the mid-IR interferometer MIDI/VLTI, which resolved the dust envelope of Sakurai's object. Results. We report the discovery of a unexpectedly compact (30 × 40 milliarcsec, 105 × 140 AU assuming a distance of 3.5 kpc), highly inclined, dust disk. We used Monte Carlo radiative-transfer simulations of a stratified disk to constrain its geometric and physical parameters, although such a model is only a rough approximation of the rapidly evolving dust structure. Even though the fits are not fully satisfactory, some useful and robust constraints can be inferred. The disk inclination is estimated to be 75° ± 3° with a large scale height of 47 ± 7 AU. The dust mass of the disk is estimated to be 6 × 10-5 M ⊙. The major axis of the disk (132° ± 3°) is aligned with an asymmetry seen in the old PN which was re-investigated as part of this study. This implies that the mechanism responsible for shaping the dust envelope surrounding Sakurai's object was already at work when the old PN formed.