We present a study of the old and intermediate-age star formation history (SFH) of the Local Group dwarf irregular galaxy (DIG) NGC 6822. It has been quantitatively obtained from the comparison of the observed [(V-I)_0_, M_I_] CM diagram with a set of model CM diagrams, computed assuming different evolutionary scenarios. The comparison has been performed through a number of indicators relative to the stars' distribution in two key structures of the CM diagram, the red-tail and the red-tangle. They contain most of the resolved stars of the galaxy which represent a major fraction of its SFH. The reliability of the model CM diagrams computed is a result of three main components: (i) a reliable set of stellar evolutionary models covering the needed range of ages and metallicities, (ii) the interpolation of the stellar evolutionary tracks of fixed mass and metallicity to determine the precise, smooth distribution of stars of any age, mass, and metallicity on the CM diagram, and (iii) a realistic simulation of the observational effects on the synthetic CM diagram. Significant constraints have been put on the SFH of NGC 6822. The main results reached are the following: (i) Star formation beginning later than 6 Gyr ago can be ruled out. Star formation beginning at about 6 Gyr ago would only be possible if the galaxy started its early star formation from gas already enriched to about half the current metallicity, or if a prompt metal enrichment occurred in the galaxy. (ii) NGC 6822 has most likely begun forming stars at a very early epoch (about 15-12 Gyr ago), from low metallicity gas. (iii) An SFR close to constant or declining in the last few Gyr of the galaxy's lifetime seems best to reproduce the observations. Short-time and small-amplitude fluctuations of the SFR may have occurred. This result solves a long lasting question about the age of the stars in the Baade's sheet, confirming, at least for NGC 6822, the original idea of Baade (1963) that DIGs contain a considerable amount of old stars.