Context. Theory surrounding the origin of the dust-laden winds from evolved stars remains mired in controversy. Characterizing the formation loci and the dust distribution within approximately the first stellar radius above the surface is crucial for understanding the physics that underlie the mass-loss phenomenon. Aims. By exploiting interferometric polarimetry, we derive the fundamental parameters that govern the dust structure at the wind base of a red supergiant. Methods. We present near-infrared aperture-masking observations of Betelgeuse in polarimetric mode obtained with the NACO/SAMPol instrument. We used both parametric models and radiative transfer simulations to predict polarimetric differential visibility data and compared them to SPHERE/ZIMPOL measurements. Results. Using a thin dust shell model, we report the discovery of a dust halo that is located at only 0.5 R_⋆ above the photosphere (i.e. an inner radius of the dust halo of 1.5 R_⋆). By fitting the data under the assumption of Mie scattering, we estimate the grain size and density for various dust species. By extrapolating to the visible wavelengths using radiative transfer simulations, we compare our model with SPHERE/ZIMPOL data and find that models based on dust mixtures that are dominated by forsterite are most favored. Such a close dusty atmosphere has profound implications for the dust formation mechanisms around red supergiants.