Context. Large cavities in disks are important testing grounds for the mechanisms proposed to drive disk evolution and dispersion, such as dynamical clearing by planets and photoevaporation. Aims. We aim to resolve the large cavity in the disk around HD 34282, whose presence has been predicted by previous studies modeling the spectral energy distribution of the disk. Methods. Using ALMA band 7 observations we studied HD 34282 with a spatial resolution of 0.10″ × 0.17′′ at 345 GHz. Results. We resolve the disk around HD 34282 into a ring between 0.24′′ and 1.15′′ (78+7-11 and 374+33-54 au adopting a distance of 325+29-47 pc). The emission in this ring shows azimuthal asymmetry centered at a radial distance of 0.46′′ and a position angle of 135° and an azimuthal FWHM of 51°. We detect CO emission both inside the disk cavity and as far out as 2.7 times the radial extent of the dust emission. Conclusions. Both the large disk cavity and the azimuthal structure in the disk around HD 34282 can be explained by the presence of a 50 Mjup brown dwarf companion at a separation of ≈0.1′′.