The charge-neutral exciton has been predicted to carry genuine photocurrent due to the geometric Berry phase of the electronic bands, if the inversion symmetry in a crystal is broken. We detect such exciton shift current in a prototypical polar semiconductor CdS by using terahertz emission spectroscopy. A distinct peak emerges in the photocurrent spectra at the energy of the exciton resonance, which is demonstrated to result from the distinct displacements of electrons and holes in real space within the excitons to produce a finite transient charge current at subpicosecond time scale. Our findings elucidate the Berry phase physics of the charge-neutral photoexcitations and also shed light on the novel energy harvesting mechanism by exciton generation.