The cross sections for charged and neutral current deep inelastic scattering in e + p collisions with a longitudinally polarised positron beam have been measured using the ZEUS detector at HERA. The results, based on data corresponding to an integrated luminosity of 23.8 pb -1 at sqrt(s) = 318 GeV, are given for both e + p charged current and neutral current deep inelastic scattering for both positive and negative values of the longitudinal polarisation of the positron beam. Single differential cross sections are presented for the kinematic region Q 2 > 200 GeV 2. The measured cross sections are compared to the predictions of the Standard Model. A fit to the data yields σ CC ( P e = - 1 ) = 7.4 ± 3.9 ( stat. ) ± 1.2 ( syst. ) pb, which is consistent within two standard deviations with the absence of right-handed charged currents in the Standard Model. © 2006 Elsevier B.V. All rights reserved.