  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 sigma^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.
