[en] The cross sections for neutral current (NC) deep inelastic scattering (DIS) in e⁺p collisions with a longitudinally polarised positron beam are measured at high momentum transfer squared (Q²>185 GeV²) at the ZEUS detector at HERA. The HERA accelerator provides e^±p collisions at a centre-of-mass energy of 318 GeV, which allows the weak contribution to the NC process to be studied at high Q². The measurements are based on a data sample with an integrated luminosity of 135.5 pb^-1 collected with the ZEUS detector in 2006 and 2007. The single differential NC cross sections dσ/dQ², dσ/dx and dσ/dy and the reduced cross section σ are measured. The structure function xF₃ is determined by combining the e⁺p NC reduced cross sections with the previously measured e^-p measurements. The interference structure function xF₃^γZ is extracted at Q²=1500 GeV². The cross-section asymmetry between the positive and negative polarisation of the positron beam is measured and the parity violation effects of the electroweak interaction are observed. The predictions of the Standard Model of particle physics agree well with the measurements. (orig.)