We study the effect of introducing hydrogen gas through the rf-plasma source during plasma-assisted molecular-beam epitaxy of GaN(0001). The well-known smooth-to-rough transition that occurs for this surface as a function of decreasing Ga flux in the absence of H is found to persist even with H present, although the critical Ga flux for this transition increases. Under Ga-rich conditions, the presence of hydrogen is found to induce step bunching (facetting) on the surface. Conductive atomic force microscopy reveals that leakage current through dislocation cores is significantly reduced when hydrogen is present during the growth.