We study the density-driven phase transition from hexagonal ice in the temperature range of 170-230 K up to pressures of 0.65 GPa using a piston-cylinder apparatus. Pure ice II, pure ice III, or mixtures of ice II and ice III are identified as products when the compression-rate is varied from 0.001-4 GPa/min. At low compression-rates and at high temperatures, formation of ice II is observed, which is in accordance with the phase diagram. However, at low temperatures and at higher compression rates, formation of metastable ice III is observed, which extends the known temperature range for possible formation of ice III. Metastable ice III rather than stable ice II crystals are produced by simple variation of the compression rate, which is an uncommon concept of producing metastable phases. We discuss some implications for our understanding of the interior of icy satellites such as Ganymede.