The ability to post-select the outcomes of an experiment is a useful theoretical concept and experimental tool. In the context of weak measurements, post-selection can lead to surprising results such as complex weak values outside the range of eigenvalues. Usually post-selection is realized by a projective measurement, which is hard to implement in ensemble systems such as NMR. We demonstrate the first experiment of a weak measurement with post-selection on an NMR quantum information processor. Our setup is used for measuring complex weak values and weak values outside the range of eigenvalues. The scheme for overcoming the problem of post-selection in an ensemble quantum computer is general and can be applied to any circuit-based implementation. This experiment paves the way for studying and exploiting post-selection and weak measurements in systems where projective measurements are hard to realize experimentally.