An electronic device may be provided with a phased antenna array that includes a dielectric resonator antenna having a dielectric column mounted to a circuit board. The dielectric column may be embedded in a dielectric substrate such as a plastic overmold. Conductive walls may be disposed on the dielectric substrate and may laterally surround the dielectric substrate and one or more dielectric resonating elements in the phased antenna array. The conductive walls may be grounded. The conductive walls may have a tapered shape. The conductive walls may help to isolate the antenna from electromagnetic influences from nearby conductive components in the electronic device. The conductive walls may form a conductive horn that helps to maximize the gain of the antenna in conveying radio-frequency signals greater than 10 GHz through a display cover layer, housing window, camera sapphire, or rear housing wall.