The dynamic spectrum of a radio pulsar is an in-line digital hologram of the ionized interstellar medium. It has previously been demonstrated that such holograms permit image reconstruction, in the sense that one can determine an approximation to the complex electric field values as a function of Doppler shift and delay, but to date the quality of the reconstructions has been poor. Here we report a substantial improvement in the method which we have achieved by simultaneous optimization of the thousands of coefficients that describe the electric field. For our test spectrum of PSR B0834+06 we find that the model provides an accurate representation of the data over the full 63 dB dynamic range of the observations: residual differences between model and data are noise like. The advent of interstellar holography enables detailed quantitative investigation of the interstellar radio-wave propagation paths for a given pulsar at each epoch of observation. We illustrate this using our test data which show the scattering material to be structured and highly anisotropic. The temporal response of the medium exhibits a scattering tail which extends to beyond 100 μs , and the centroid of the pulse at this frequency and this epoch of observation is delayed by approximately 15 μs as a result of multipath propagation in the interstellar medium.