We study the formation of disc galaxies in a fully cosmological framework using adaptive mesh refinement simulations. We perform an extensive parameter study of the main subgrid processes that control how gas is converted into stars and the coupled effect of supernovae feedback. We argue that previous attempts to form disc galaxies have been unsuccessful because of the universal adoption of strong feedback combined with high star formation efficiencies. Unless extreme amounts of energy are injected into the interstellar medium during supernovae events, these star formation parameters result in bulge-dominated S0/Sa galaxies as star formation is too efficient at z ~ 3. We show that a low efficiency of star formation more closely models the subparsec physical processes, especially at high redshift. We highlight the successful formation of extended disc galaxies with scalelengths rd = 4-5 kpc, flat rotation curves and bulge-to-disc ratios of B/D ~ 1/4. Not only do we resolve the formation of a Milky Way-like spiral galaxy, we also observe the secular evolution of the disc as it forms a pseudo-bulge. The disc properties agree well with observations and are compatible with the photometric and baryonic Tully-Fisher relations, the ΣSFR -Σgas (Kennicutt-Schmidt) relation and the observed angular momentum content of spiral galaxies. We conclude that the underlying small-scale star formation physics plays a greater role than previously considered in simulations of galaxy formation.