We report on the growth of GaN quantum dots and the control of their density in the Stranski-Krastanov mode on AlN (0001) by rf-plasma molecular beam epitaxy at 750 degreesC. After depositing the equivalent of 2-3 ML GaN coverage, as limited by N fluence under Ga-droplet growth conditions, excess Ga was desorbed and Stranski-Krastanov islands formed under vacuum. We present the dependence of island density as a function of GaN coverage (for two growth rates: 0.10 and 0.23 ML/s), as estimated from atomic force microscopy and cross-sectional transmission electron microscopy. With a GaN growth rate of 0.23 ML/s, the island density was found to vary from less than 3.0x10(8)-9.2x10(10) cm(-2) as the GaN coverage was varied from 2.2 (critical thickness) to 3.0 ML. For a GaN growth rate of 0.10 ML/s, the island density varied from 2.0x10(10) to 7.0x10(10) cm(-2) over a GaN coverage range of 2.0-3.0 ML. For each growth rate, the GaN islands were found to be of nearly uniform size, independent of the quantum dot density. (C) 2004 American Institute of Physics.