We propose and provide experimental evidence in support of a theory for the remote preparation of a complex spatial state of a single photon. An entangled two-photon source was obtained by spontaneous parametric down-conversion, and a double slit was placed in the path of the signal photon as a scattering object. The signal photon was detected after proper spatial filtering so that the idler photon was prepared in the corresponding single-photon state. By using a two-photon coincidence measurement, we obtained the Radon transform, at several longitudinal distances, of the single-photon Wigner distribution function modified by the double slit. The experimental results are consistent with the idler photon being in a pure state. An inverse Radon transformation can, in principle, be applied to the measured data to reconstruct the modified single-photon Wigner function, which is a complete representation of the amplitude and phase structure of the scattering object.