The production of Z0 bosons in the reaction ep -> eZ0p*, where p* stands for a proton or a low-mass nucleon resonance, has been studied in ep collisions at HERA using the ZEUS detector. The analysis is based on a data sample collected between 1996 and 2007, amounting to 496 pb-1 of integrated luminosity. The Z0 was measured in the hadronic decay mode. The elasticity of the events was ensured by a cut on eta_max < 3.0, where eta_max is the maximum pseudorapidity of energy deposits in the calorimeter defined with respect to the proton beam direction. A signal was observed at the Z0 mass. The cross section of the reaction ep -> eZ0p* was measured to be sigma(ep -> eZ0p*) = 0.13 +/- 0.06 (stat.) +/- 0.01 (syst.) pb, in agreement with the Standard Model prediction of 0.16 pb. This is the first measurement of Z0 production in ep collisions.