We present a mechanism to produce indistinguishable single-photon pulses on demand from a single atom-optical cavity system. We use sequences of two laser pulses of alternate circular polarizations at the two Raman transitions of a four-level atom. They allow the production of the same cavity-mode photons without repumping of the atom between photon generations. Photons that are emitted from the cavity with near-unity efficiency in well-defined temporal modes, feature the same polarization, frequency and identical shapes, controlled by the laser fields. The second order correlation function reveals the single-photon nature of the proposed source. A realistic setup for the experimental implementation is presented.