We report the current-perpendicular-to-plane giant magnetoresistance of a spin valve with Co2MnSi (CMS) Heusler alloy ferromagnetic electrodes. A multilayer stack of Cr/Ag/Cr/CMS/Cu/CMS/Fe25Co75/Ir28Mn72/Ru was deposited on a MgO (001) single crystal substrate. The bottom CMS layer was epitaxially grown on the Cr/Ag/Cr buffer layers and was ordered to the L21 structure after annealing at 673 K. The upper CMS layer was found to grow epitaxially on the Cu spacer layer despite the large lattice mismatch between Cu and CMS. The highest MR ratios of 8.6% and 30.7% for CPP-GMR were recorded at room temperature and 6 K, respectively. The high spin polarization of the epitaxial CMS layers is the most likely origin of the high MR ratio.