Abstract submitted to the XXXth International Conference on High Energy Physics July 27 - August 2, 2000, Osaka, Japan _________________________________________________________________________ Measurement of azimuthal asymmetries in deep inelastic scattering ZEUS Collaboration Abstract The distribution of the azimuthal angle for charged hadrons has been studied in the hadronic centre-of-mass system for neutral current deep inelastic positron-proton scattering with the ZEUS detector at HERA, using an integrated luminosity of 38~pb$^{-1}$. Measurements of the dependence of the moments of this distribution on the transverse momenta of the charged hadrons are presented. Asymmetries that can be unambiguously attributed to perturbative QCD processes have been observed for the first time.