Zinc molybdate (ZnMoO4) single crystals were grown for the first time by the Czochralski method and their luminescence was measured under X ray excitation in the temperature range 85-400 K. Properties of ZnMoO4 crystal as cryogenic low temperature scintillator were checked for the first time. Radioactive contamination of the ZnMoO4 crystal was estimated as <0.3 mBq/kg (228-Th) and 8 mBq/kg (226-Ra). Thanks to the simultaneous measurement of the scintillation light and the phonon signal, the alpha particles can be discriminated from the gamma/beta interactions, making this compound extremely promising for the search of neutrinoless Double Beta Decay of 100-Mo. We also report on the ability to discriminate the alpha-induced background without the light measurement, thanks to a different shape of the thermal signal that characterizes gamma/beta and alpha particle interactions.