Epitaxial growth of Cu/Nb multilayer on alpha-Al2O3 was realized under UHV electron beam evaporation by controlling the substrate temperature at each stage of multilayer growth. The typical thicknesses of Cu and Nb layers are in the range of 50 to 260 nm. The crystal quality except the interface boundary is shown to be high with RBS/channeling and XRD analyses. It has been found that the deposited Cu and Nb layers are single-crystalline over the sample size far beyond the TEM scale. The orientation relationship between Cu and Nb layers is as follows: Cu(111)/Nb(110) on alpha-Al2O3 n-plane, Cu(001)/Nb(001) on alpha-Al2O3 r-plane and textured Cu(111)/Nb(111) on alpha-Al2O3 c-plane. The suitable temperature for single-crystalline Nb growth is 750 degrees C in common with three different alpha-Al2O3 substrate and those for single crystalline Cu growth on Nb(110) and Nb(001) layers are 200 degrees C and RT, respectively. It is concluded that Cu(111)/Nb(110) with Cu[1(1) over bar 0]//Nb[001] is crystallographically consistent stacking in a hetero-epitaxial manner.