We report on hexagonal aluminum nitride (h-AlN) crystal growth in the Al-N system by means of AlN recrystallization in the field of a temperature gradient at 6 GPa and 1800 degrees C. A special setup for h-AlN growth using a multianvil apparatus has been developed, which allows the growth of colorless transparent h-AlN single crystals. Crystals exhibited two distinct morphological types. The first is short prismatic and platelet-like crystals with smooth faces and a size up to 0.4 mm. The second is skeleton crystals elongated along the [1210] direction and flattened along the (0001) plane with a size of up to 1.0 mm. Crystals have been characterized by scanning electron microscopy, Raman spectroscopy, and electron backscattered diffraction methods. The growth mechanism and main factors determining the crystal habit and growth rate in a high-pressure solution method are discussed. We also discuss the possibility of using the employed technique for cubic aluminum nitride (c-AlN) crystal growth.