We aim to provide here the first data-constrained estimate of the metal-free (Population III; Pop III) star formation rate density rho(III)(*) required at high redshifts (z greater than or similar to 16) in order to reproduce both the amplitude and the redshift of the EDGES 21-cm global signal. Our model accounts for the Ly alpha, radio, and X-ray backgrounds from both Pop III and metal-enriched Population II (Pop II) stars. For the latter, we use the star formation rate density estimates (and the Ly alpha background) from the Delphi semi-analytic model that has been shown to reproduce all key observables for galaxies at z greater than or similar to 5; the radio and X-ray backgrounds are fixed using low-z values. The constraints on the free parameters characterizing the properties of the Pop III stars are obtained using a Markov Chain Monte Carlo analysis. Our results yield rho(III)(*) that while increasing from z similar to 21 to 16 thereafter shows a sharp decline which is in excellent agreement with the results found by Valiante et al. to simulate the growth of z similar to 6-7 quasars and their host galaxies, suggesting that the bulk of Pop III star formation occurs in the rarest and most massive metal-poor haloes at z less than or similar to 20. This allows Pop III stars to produce a rapidly growing Ly alpha background between z similar to 21 and 15. Further, Pop III stars are required to provide a radio background that is about 3-4 orders of magnitude higher than that provided by Pop II stars although Pop II stars dominate the X-ray background.