The numerous neurons and glia that form the brain originate from tightly controlled growth and division of neural stem cells, regulated systemically by important known stem cell-extrinsic signals. However, the cell-intrinsic mechanisms that control the distinctive proliferation rates of individual neural stem cells are unknown. Here, we show that the size and division rates of Drosophila neural stem cells (neuroblasts) are controlled by the highly conserved RNA binding protein Imp (IGF2BP), via one of its top binding targets in the brain, myc mRNA. We show that Imp stabilises myc mRNA leading to increased Myc protein levels, larger neuroblasts, and faster division rates. Declining Imp levels throughout development limit myc mRNA stability to restrain neuroblast growth and division, and heterogeneous Imp expression correlates with myc mRNA stability between individual neuroblasts in the brain. We propose that Imp-dependent regulation of myc mRNA stability fine-tunes individual neural stem cell proliferation rates.