Infrared spectroscopy has been extensively used to determine the mineralogy of circumstellar dust. The identification of dust species with featureless opacities, however, is still ambiguous. Here we present a method to lift the degeneracy using the combination of infrared spectroscopy and interferometry. Aims. The binary post-AGB star HR 4049 is surrounded by a circumbinary disk viewed at a high inclination angle. Apart from gaseous emission lines and molecular emission bands of polycyclic aromatic hydrocarbons (PAH), diamonds, and fullerenes, the 2–25 μm infrared spectrum is featureless. The goal of the paper is to identify the dust species responsible for the smooth spectrum. Methods. We gathered high-angular-resolution measurements in the near- and mid-infrared with the VLTI interferometric instruments AMBER and MIDI. The data set is expanded with archival Geneva optical photometry, ISO-SWS and Spitzer-IRS infrared spectroscopy, and VISIR N-band images and spectroscopy. We computed a grid of radiative-transfer models of the circumbinary disk of HR 4049 using the radiative-transfer code MCMax. We searched for models that provide good fits simultaneously to all available observations. Results. We find that the variable optical extinction towards the primary star is consistent with the presence of very small (0.01 μm) iron-bearing dust grains or amorphous carbon grains. The combination of the interferometric constraint on the disk extent and the shape of the infrared spectrum points to amorphous carbon as the dominant source of opacity in the circumbinary disk of HR 4049. The disk is optically thick to the stellar radiation in the radial direction. At infrared wavelengths it is optically thin. The PAH emission is spatially resolved in the VISIR data and emanates from a region with an extent of several hundreds of AU, with a projected photocenter displacement of several tens of AU from the disk center. The PAHs most likely reside in a bipolar outflow. Conclusions. Dust species with featureless opacity curves, such as metallic iron and amorphous carbon, can be identified by combining infrared spectroscopy and high-angular-resolution measurements. In essence, this is because the temperatures of the dust species are notably different at the same physical distance to the star.