The $e^+p$ neutral-current deep inelastic scattering differential cross-sections $d\sigma/dQ^2$, for $Q^2$ > 400 $GeV^2$, $d\sigma/dx$ and $d\sigma/dy$, for $Q^2$ > 400, 2500 and 10000 $GeV^2$, have been measured with the ZEUS detector at HERA. The data sample of $47.7 pb^{-1}$ was collected at a center-of-mass energy of 300 GeV. The cross-section, $d\sigma/dQ^2$, falls by six orders of magnitude between $Q^2 = 400$ and $40000 GeV^2$. The predictions of the Standard Model are in very good agreement with the data. Complementing the observations of time-like $Z^0$ contributions to fermion-antifermion annihilation, the data provide direct evidence for the presence of $Z^0$ exchange in the space-like region explored by deep inelastic scattering.