The production of Z0Z0 bosons in the reaction ep→eZ0p(⁎)ep→eZ0p(⁎), where p(⁎)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−1496 pb−1 of integrated luminosity. The Z0Z0 was measured in the hadronic decay mode. The elasticity of the events was ensured by a cut on ηmax<3.0ηmax<3.0, where ηmaxηmax is the maximum pseudorapidity of energy deposits in the calorimeter defined with respect to the proton beam direction. A signal was observed at the Z0Z0 mass. The cross section of the reaction ep→eZ0p(⁎)ep→eZ0p(⁎) was measured to be σ(ep→eZ0p(⁎))=0.13±0.06(stat.)±0.01(syst.) pbσ(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 Z0Z0 production in ep collisions.