Binding energy of the 1s exciton state in the Zn1-xCdxO/ZnO quantum wells, changing between 10 and 50 angstrom in width, has been calculated for x up to 0.2. The exciton wave function is represented in terms of hydrogenic wave functions and the effects of dielectric mismatch, exciton-phonon interactions and built-in electric field on the binding energy are considered. The calculated built-in electric field is around 19.6x MV cm(-1) in a 20-angstrom Zn1-xCdxO/ZnO quantum well. The numerical results indicate that the exciton binding energy stays below its bulk value virtually for all x because of the built-in electric field dominating the quantum confinement effect. The binding energy is nearly 26 meV at maximum, barely above its bulk value, and about 13 meV at minimum. The contribution of the dielectric mismatch is up to 40% while that of the exciton-phonon interactions is negligible. The exciton binding energy increases up to 66 meV, well above the bulk value, in case the built-in electric field is neglected. A qualitative agreement between the calculated excitonic transition energies and the low temperature and low excitation intensity photoluminescence peak data is obtained although some ZnCdO material properties, such as low temperature band gap, are not available.