Measurements of neutral current cross sections for deep inelastic scattering in e+p collisions at HERA with a longitudinally polarised positron beam are presented. The single-differential cross-sections dσ/dQ2, dσ/dx and dσ/dy and the reduced cross-section ˜σ were measured in the kinematic region Q2 > 185GeV2 and y < 0.9, where Q2 is the four-momentum transfer squared, x the Bjorken scaling variable, and y the inelasticity of the interaction. The measurements were performed separately for positively and negatively polarised positron beams. The measurements are based on an integrated luminosity of 135.5 pb−1 collected with the ZEUS detector in 2006 and 2007 at a centre-of-mass energy of 318GeV. The structure functions F˜3 and F Z 3 were determined by combining the e+p results presented in this paper with previously published e−p neutral current results. The asymmetry parameter A+ is used to demonstrate the parity violation predicted in electroweak interactions. The measurements are well described by the predictions of the Standard Model.