IceCube, a future km³ antarctic ice Cherenkov neutrino telescope, is highly sensitive to a galactic supernova (SN) neutrino burst. The Cherenkov light corresponding to the total energy deposited by the SN neutrinos in the ice can be measured relative to background fluctuations with a statistical precision much better than 1%. If the SN is viewed through the Earth, the matter effect on neutrino oscillations can change the signal by more than 5%, depending on the flavour-dependent source spectra and the neutrino mixing parameters. Therefore, IceCube together with another high-statistics experiment like Hyper-Kamiokande can detect the Earth effect, an observation that would identify specific neutrino mixing scenarios that are difficult to pin down with long-baseline experiments. In particular, the normal mass hierarchy can be clearly detected if the third mixing angle is not too small, sin²θ₁₃10⁻³. The small flavour-dependent differences of the SN neutrino fluxes and spectra that are found in state-of-the-art simulations suffice for this purpose. Although the absolute calibration uncertainty at IceCube may exceed 5%, the Earth effect would typically vary by a large amount over the duration of the SN signal, obviating the need for a precise calibration. Therefore, IceCube with its unique geographic location and expected longevity can play a decisive role as a `co-detector' to measure SN neutrino oscillations. It is also a powerful stand-alone SN detector that can verify the delayed-explosion scenario.